Did Lucian of Samosata make this joke on purpose?

No. Lukian did not make that joke on purpose.

A relatively modern translator made this deadpan translation most probably without it being intended as any form of joke, since the context seems otherwise devoid of any such hints to suggest such an interpretation.

The vast majority of translations into modern languages do not make this choice. Sexual innuendo of any kind is not part of either Greek or most translations of this short passage. It makes not much sense to read this alleged double-entendre into it.

Another, more recent translation goes like this:

H. I’ll go up again. Good: I can see everything. Up you come now.

C. Give me a hand, Hermes: this is a monstrous contrivance you are making me climb.

— C. D. N. Costa: "Lucian", Oxford World Classics, Oxford University Press: Oxford, New York, 2005.

While the Loeb Classical Library renders it:

Hermes
Never fear; everything will be secure. Move Oeta over. Roll Parnassus this way. There now, I am going up again. It is all right, I see everything; now come up yourself.

Charon
Put out your hand, Hermes. This is an uncommonly big piece of stage-machinery that you are mounting me on.

— A M Harmon (ed, tr): "Lucian", Vol 2, Heinemann, London, New York, 1919.

Another translation from 2017 reads as:

Give me a hand, Hermes, this is some contraption you want me to climb up.

— Lionel Casson: "Selected Satires of Lucian", Routledge: London, New York, 2017. p 221, gBooks

And many more to be found on Lucian of Samosata Project.

The translation word choice for 'erection' appears to be from T. R. Mills, version of "Lucian: Charon and Timon", London: W. B. Clive, University Tutorial Press, (no date?). link Which has ample notes but seems to refuse to explain this peculiar word choice?

The original Greek reads:

ΧΑΡΩΝ Ὄρεξον, ὦ Ἑρμῆ, τὴν χεῖρα· οὐ γὰρ ἐπὶ μικράν με ταύτην μηχανὴν ἀναβιβάζεις.

A dictionary for the word μηχανή explains:

μηχα^ν-ή , Dor. μα_χα^νά , ἡ, (μῆχος):
I. contrivance, esp. machine for lifting weights and the like , crane, Hdt.2.125, IG11(2).161 A69, al. (Delos, iii B. C.); μ. τετράκωλος, δίκωλος, Rev.Phil.44.251 (Didyma, ii B. C.); “μ. λιθαγωγός” Poll. 10.148; ἰχθυβόλῳ μ., of Poseidon's trident, A.Th.132 (lyr.); λαοπόροις μ., of Xerxes' bridge of boats, Id.Pers.114 (lyr.), cf. 722; freq. of irrigation machines, POxy.985 (i A. D.), etc.; also of oil-presses, Wilcken Chr.176.10 (i/ii A. D.), etc. 2. engine of war, “μηχανὰς προσάγειν” Th.2.76, etc.; “ἑλεῖν μηχαναῖς” Id.4.13. 3. theatrical machine by which gods, etc., were made to appear in the air, Pl.Cra.425d, Clit.407a; “αἴρειν μ.” Antiph.191.15, Alex.126.19: hence, prov. of anything sudden and unexpected, “ἀπὸ μηχανῆς θεὸς ἐπεφάνης” Men. 227; “ὥσπερ ἀπὸ μ.” D.40.59, cf. Arist.Po.1454b1. 4. area of land irrigated by a machine, POxy.1830.13 (vi A. D.), PLond.5.1765 (vi A. D.), PSI1.77.14 (vi A. D.). II. any artificial means or contrivance for doing a thing, “ἤτοι κλήρῳ . . , ἢ ἄλλῃ τινὶ μ.” Hdt.3.83; “εἴ τίς ἐστι μ., ἴθι καὶ πειρῶ” Id.8.57, etc.; esp. in pl. μηχαναί, shifts, devices, wiles, Hes.Th. 146; “πάντα σοφίσματα καὶ πάσας μ. ἐπεποιήκεε ἐς αὐτοὺς Δαρεῖος” Hdt. 3.152; μηχαναῖς Διός by the arts of Zeus, A.Ag.677; χερὸς . . ἐκτίνοντα μηχανάς acts of violence, ib.1582; “Ὀρέστην μηχαναῖσι μὲν θανόντα, νῦν δὲ μηχαναῖς σεσωσμένον” S.El.1228; “κρατεῖ μαχαναῖς . . θηρός” Id.Ant.349 (lyr.); “σοφιστῶν μ.” Pl.Lg.908d: prov., “μηχαναὶ Σισύφου” Ar.Ach.391:—Phrases: “πάσας προσφέροντε μ.” E.IT112; “μηχανὴν προσοιστέον” Ar.Th.1132; “πᾶσαν σπουδὴν καὶ μ. προσφερόμενος” Plb.1.18.11; “ἐπεισήγαγον μ.” Id.29.25.1; “μηχανὰς εὑρήσομεν, ὥστε ἀπαλλάξαι” A.Eu.82; “πλέκειν” E.Andr.66; “πορίζεσθαι” Pl.Smp. 191b; “ἐκπορίζειν” Ar.V.365; ζητεῖν ib.149; ἀντλεῖν μαχανάν exhaust one's resources, Pi.P.3.62; κατ᾽ ἐμὰν μ. ib.109: c. gen. objecti, ἔξευρε μ. τιν᾽ Ἀδμήτῳ κακῶν contrivance against ills, E.Alc.221 (lyr.); but μ. σωτηρίας a way, means of procuring or providing safety, A. Th.209; “μυρίων οὐσῶν μ. ἀπαλλαγῆς” X.Cyr.5.1.12; οὐδεμία μ. [ἐστι] ὅκως οὐ c. fut. ind., Hdt.2.160; μὴ οὐ c. inf., ib. 181, 3.51; τὸ μὴ οὐ (prob.) Id.1.209; τίς μ. μὴ οὐχὶ . . ; Pl.Phd.72d. 2. freq. in adverb. phrases, μηδεμιῇ μ. by no means whatsoever, by no contrivance, Hdt. 7.51, etc.; “οὓς οὐδεμιῇ μ. δεῖ τιτρώσκειν” Hp.Art.11; so μήτε τέχνῃ μήτε μ. μηδεμιᾷ Foed. ap. Th.5.18, cf. IG12.39.23; opp. “πάσῃ τέχνῃ καὶ μ.” Lys. 19.53; “πάσῃ μ.” Ar.Lys.300 (lyr.); τρόπῳ ἢ μ

Now, in English we see for the chosen word 'erection' the following very first entry in OED:

erection
b. concr. That which is built up or reared; a building, structure. lit. and fig.

So, an actual contention seems to arise that in this one peculiar translation presented in the question also ἀναβιβάζεις was expressed as 'to mount', apparently read by some as too closely connected to the 'erection' as well. At this point, note that the rather isolated translation inquired about also chooses "and no mistake" for some reason…

But ἀναβιβάζεις is again primarily 'make to go up':

ἀναβιβάζω Causal of ἀναβαίνω.

I.to make go up, cause to mount, ἐπὶ πύργον Hdt., Xen. II.Special usages: 1.ἀν. τινα ἐφ᾽ ἵππον to mount one on horseback, id=Xen.; ἐπ᾽ ἅρμα Hdt. 2.ἀν. ναῦν to draw a ship up on land, Xen. 3.Mid., ἀναβιβάζεσθαί τινας ἐπὶ τὰς ναῦς to have them put on board ship, Thuc.; absol., ἀναβιβασάμενοι id=Thuc. 4.at Athens, to bring up to the bar as a witness, Plat., etc.: Mid., of a culprit, to bring up his wife and children to raise compassion, id=Plat. 5.Pass., ἀναβιβάζεσθαι εἰς τιμήν to ascend to honour, Plut. 6.ἀν. τοὺς φθόγγους, to moderate them, id=Plut.

Comparing the dictionary entries for the two peculiar word choices from Ancient Greek to modern English (or French, German…), there is hardly a hint for any possible innuendo in its usage. The surrounding text in Lucian's Charon Contemplates gives no indication or opportunity for this. No classicist seems to have written any paper describing this passage as 'saucy'. Compare for example the following treatise that lists words used for the assumed purpose, admits that common dictionaries are a bit reluctant to explore these lower registers of word meanings, lists numerous examples, also from Lucian, but not one from this passage or even this dialogue:

There existed in Greek a multitude of words denoting or connoting sexual congress. The list of verbs given by Pollux (5.92) only skims the surface. In what follows I discuss words which with one exception are absent from this list and belong, as will be seen from their distribution, to the lower register of the Greek language. They are all demonstrably direct expressions, blunt and non-euphemistic.

— David Bain: "Six Greek Verbs of Sexual Congress (βινω̑, κινω̑, πυγίζω, ληκω̑, οἴφω, λαικάζω)", The Classical Quarterly , 1991, Vol. 41, No. 1 (1991), pp. 51–77. jstor

Most other translations into English seem to choose words very differently, so that the question as posed does not present itself. The first translation shown in this answer is the most modern translation, and is devoid of sexual connotations. As the body text in the question itself exemplifies: Charon and Hermes "collect a variety of mountains and cobble them together to erect one single big mountain." A sentence in English that features the strange word but does not lend itself so readily to reading sexual content into it, despite the whole question being about exactly such possibilities and thus setting up a certain nudging frame for interpreting such words.

To the contrary, the recent French edition of Lucian has for Charon Ou Les Contemplateurs the translation:

HERMÈS. — N’aie pas peur, il n’y aura aucun danger. Transporte-moi l’Œta et roulons dessus le Parnasse. Vois, à présent je remonte. C’est parfait. Je vois le monde entier. Monte maintenant, toi aussi.

CHARON. — Tends-moi la main, Hermès ; car ce n’est pas sur une petite machine¹⁷ que tu me fais monter ici.

— Èmile Chambry, Alain Billault, Èmeline Marquis, Domic Goust: "Lucien de Samosate. Œuvres complètes", Robert Laffont: Paris, 2015

With footnote 17 explaining:

17. Allusion à la méchané du théâtre (il s’agit d’un treuil élévateur).

German translations also avoid any of the saucy parts:

Gib mir die Hand, Hermes; es ist kein kleines Gerüst, worauf du mich steigen lässt. (Give me your hand, Hermes; it is not a small scaffold on which you let me ascend.)

Or a similar scaffold:

Reiche mir die Hand, Merkur! Es ist ein ansehnliches Gerüstchen, das ich besteigen soll.

Keeping in mind that Charon is saying in Greek in the passive that there would be 'an erection' the 'he is made 'to mount' by Hermes', the 'physical/mechanical' workings of such a supposed constellation seem a bit implausible as well. Problems of picturing it disappear when it is read straight: as Hermes making Charon go up an artificial mountain with Charon commenting on that.

The direct and simple meaning is therefore captured in this interpretation:

There is a doublet in Charon: Hermes has to take Pelion, Parnassus and Oeta back down from the top of Olympus. […]

Hermes finds his old companion Charon lost in an equally unexpected place (above ground), and also finds a quick way to show him the sights of this new world (4-7). Here we find Solon introduced ex machina, like Philosophy herself in Piscator, and in almost the same words. […]

or they transport whole districts in the same way as Lucian's gods move mountains, […]

In Charon 4 Hermes wishes to show Charon a marvellous sight: he hits on the plan of calling in Homer to perform a miracle and put mountains together to make the spectacle possible. […]

In both passages a cautious friend intervenes and tries to opt out of such a mad scheme before it is too late: Charon feels that the mountains are a rather precarious stage-set; […, footnoted with: cf. Trygaeus' instructions to the scene-shifters at Ar. Pax. 174 ff.; Navig. 30.]

— Graham Anderson: "Lucian. Theme and Variation In The Second Sophistic", Brill: Leiden, 1976.

The original text reads

Ὄρεξον, ὦ Ἑρμῆ, τὴν χεῖρα· οὐ γὰρ ἐπὶ μικράν με ταύτην μηχανὴν ἀναβιβάζεις.
Órexon, ó Ermi, tin cheira: ou gár epi mikrán me táutin mechanin anabibázeis.

With the help of Eulexis, I get the following literal translation for each word:

ὄρεξον - reach (imperat act 2nd sg)
ὦ - oh
Ἑρμῆ - Hermes
τὴν - your
χεῖρα - hand (fem acc sg)
οὐ - in truth
γὰρ - for
ἔπι - being upon
μικράν - small (fem acc sg)
με - I at least (acc 1st sg)
ταύτην - this (fem acc sg)
μηχανὴν - contrivance, mechanism (fem acc sg)
ἀναβιβάζεις - make to mount (pres ind act 2nd sg)

The best literal translation I can think of is:

Give me your hand, Hermes, for I am small to be upon this contrivance you are mounting me on.

To give this a sexual conotation, the joke would have to be that Charon is about to "mount" a machine that is too large for him. That sounds...hurtfull (and very rude).

As LangLangC cited, mechani can also be a stage machine, so the joke in the original could also be that Charon hints that the "heap of mountains" is, in reality, just a stage prop, an illusion. This would fit with the means they use to lift the mountains: poetry. As is described beforehand in detail, Hermes got the idea for this from Homer's Odyssey (Book 11, 315), where two brothers try to scale the heavens by stacking mountains. "He made it forthtwith to scale Heaven with a pair of verses" (“Ομηρος ἀπὸ δυοῖν στί- χοιν αὐτίκα ἡμῖν ἀμβατὸν ἐ ἐποίησε τὸν οὐρανόν, οὕτω ῥᾳδίως συνθεὶς τὰ ὄρη.) - The wonderous power of the word turns impossible work into the wonderous magic of the stage: imaginary, but powerful.