What is this ancient Egyptian material called "maklalu"?

There a few aspects to consider here:
An older etymological study identified maklulu as a Sumerian loanword entering the Akkadian language, meaning (or at least being very closely related to) a body part: the female womb. (Harri Holma: "Die Namen der Körperteile im Assyrisch-Babylonischen, eine lexikalisch-etymologische Studie", Leipzig, 1911, p 109.)

maklulu (a garment) CAD M/i, 137b s.v. ‘maklalu’ This is an article of clothing, better attested in Neo-Assyrian texts. In MARV 1.24:7’ it appears to be ‘for work’ (ša kin), and to be of some technique or workmanship which is lost at the broken end of the line. In KAV 99 from the Babu-aḫa-iddina archive we have “two work garments of the countryside, together with their maklulu”, suggesting that it could be an ancillary item, conceivably ‘hood’ (given the association of the root kll with headgear in other words).

Nicholas Postgate: "Wool, Hair and Textiles in Assyria", in: Catherine Breniquet and Cécile Michel (Eds.): "Wool Economy In The Ancient Near East And The Aegean. From the Beginnings of Sheep Husbandry to Institutional Textile Industry", Oxbow Books: Oxford, Philadelphia, 2014, p 401– 427.

But:

Assyrian designations belonging to the common Akkadian textile vocabulary:
maklulu or muklālu (muqlālu). This term, derived from the verb qalālu, ‘to be light, weak’, seems to denote a wool shawl or a cape. In a Middle Assyrian text wool garments (lubēru) with their maklalu are listed. The textiles in question are qualified as garments ša ṣēri, ‘of the steppe/countryside’, perhaps, to be intended as garments with capes which were used for travel or which were characteristic of the nomads’ dress. Postgate suggests the translation ‘hood’. Moreover, it seems that in 2nd-millennium BC Assyria also maklulus for work (ša šipri/KIN) were in use. The Neo-Assyrian maklulu came in two varieties: one with sleeves and one without sleeves. Administrative texts dealing with textiles tell us that the muklālu could be made of biršu, and that it could have a red coloured front-piece and (precious) stones sewn onto it, perhaps along the border. Another document specifies that the colour used for the front-part of the muklālu was the commercial red.

Salvatore Gaspa: "Garments, Parts of Garments, and Textile Techniques in the Assyrian Terminology: The Neo-Assyrian Textile Lexicon in the 1st-Millennium BC Linguistic Context" (2017). Textile Terminologies om the Orient to the Mediterranean and Europe, 1000 BC to 1000 AD. 3. http://digitalcommons.unl.edu/texterm/3

Leaving us to conclude that this probably was not really an Egyptian garment, at least not one 'invented there'. Probably copied from the Assyrian/Akkadian style or imported at a premium from there. But the words seem to be used as internationally defined terms to facilitate trade, being identical and equally well understood in Eygptian, Akkadian, Hittite markets.

From the passage cited it seems to imply that the material used would be linen. But together with the assumption that it might mean hood/hooded these two assumptions are both not entirely clear. First the word for hood is listed separately in a letter as parsigu (AHw II 836). Second the material used for it might actually be linen, wool, or byssos. Conversely the term listed as one possible material was "birsu; pi. m. & f. (a coarse fabric, phps.) 'felt'."

This implies much stronger connection to the weaving technique, style of cut or intended use. Small hints from middle Assyrian texts seem to point primarily into the direction of 'travel clothes'. (Elmar Edel: "Ägyptische Ärzte und ägyptische Medizin am hethitischen Königshof. Neue Funde von Keilschriftbriefen Ramses' II. aus Bogazköy", Westdeutscher Verlag: Göttingen, 1976, p 1976.)

Since most of the time the maklalu material is actually listed as "of colourful fine linen" or "of kingly linen" (Cf James Michael Burgin: "Aspects Of Religious Administration In The Hittite Late New Kingdom", Dissertation, University of Chicago, 2016.PDF) this suggests even more so that the material origin itself is not meant but another aspect of the garment to be furnished from that, with different levels of quality available.

Others, however, refer to the position of the textile on the body and/or are in association with other items of clothing (elītu, ša muhhi, ša qabli, and šupālītu). Others may possibly be connected to their workmanship (maklulu, ‘the light one?’). Some visual characteristics of the end product, such as the ša taluk ṣirri, probably indicate the use of a finely-woven fabric, which generated an undulating movement when its wearer walked. (Gaspa 2017)

In conclusion we might assume that maklalus in that letter were some kind of cape, usually dyed red or blue, of a comparatively light material of various origins, in a style of "barbarous" or foreign origins. Its name maklalus coming from Sumerian/Akkadian language but in the international trade network of the Eastern Mediterranean coming to denote one specialised and recognisable type of clothing. "Made for work" if intended for lower classes and "made for travel" in the case of higher classes.