American European living in UK. Lost EU passport. Can I travel on American passport within EU?

First, as noted by MJeffryes in a helpful comment, you can use an EU national ID card instead of an EU passport to enter the UK, if you have one. This is fundamental in the EU's free movement directive (2004/38/EC) and the UK's implementing Immigration (EEA) Regulations.

The relevant part of the directive is Article 5(1):

Article 5

Right of entry

1. Without prejudice to the provisions on travel documents applicable to national border controls, Member States shall grant Union citizens leave to enter their territory with a valid identity card or passport....

(emphasis added)

The relevant part of the regulations is rule 11(1):

PART 2
EEA RIGHTS

Right of admission to the United Kingdom
11.—(1) An EEA national must be admitted to the United Kingdom on arrival if the EEA national produces a valid national identity card or passport issued by an EEA State.

(emphasis added)

Second, you can enter as an EU citizen without an EU passport by invoking the provision in both the directive and the regulations that allows you to prove "by other means" that you are entitled to do so.

The relevant part of the directive is Article 5(4):

4. Where a Union citizen, or a family member who is not a national of a Member State, does not have the necessary travel documents or, if required, the necessary visas, the Member State concerned shall, before turning them back, give such persons every reasonable opportunity to obtain the necessary documents or have them brought to them within a reasonable period of time or to corroborate or prove by other means that they are covered by the right of free movement and residence.

(emphasis added)

The relevant part of the regulations is rule 11(4):

(4) Before an immigration officer refuses admission to the United Kingdom to a person under this regulation because the person does not produce on arrival a document mentioned in paragraph (1) or (2), the immigration officer must provide every reasonable opportunity for the document to be obtained by, or brought to, the person or allow the person to prove by other means that the person is—

(a) an EEA national;

...

(emphasis added)

Now I do not know what "other means" might be sufficient, so it might be more expedient for you just to enter as a visa-free visitor. In fact that might be problematic, because one requirement for that is to be a "genuine visitor," which you most certainly are not. But suppose you are in fact granted leave to enter as such. That leads to the next point.

Third, even if you enter as a "third-country" national visitor, you still enjoy the rights of an EU citizen because you still are an EU citizen. This is a good thing, because if it were not true, you would be obliged not only to leave the UK within six months, but also to refrain from working while you were in the UK. Most people in your situation will be relieved to find out that they are not required to take a leave of absence from their employment until they can get a new passport or ID card. This is made explicit in the regulations in the first paragraph of Schedule 3:

SCHEDULE 3
EFFECT ON OTHER LEGISLATION

Leave under the 1971 Act
1. Where a person has leave to enter or remain under the 1971 Act which is subject to conditions and that person also has a right to reside under these Regulations, those conditions do not have effect for as long as the person has that right to reside.

(emphasis added)

So, to answer your questions explicitly:

My concern is that re-entering the UK with my american passport will log me into the system as a tourist instead as my current resident status.

That might happen, but no ill will come of it. It's merely an administrative concern. If anyone suggests that you've overstayed, you can just tell them that you're an EU citizen.

Does this mean I need to re-enter the UK when I get my EU passport back within the duration of the implicit US tourist visa?

No.

A final caveat: the UK is currently slated to leave the EU on January 31st, 2020. The explicit precedence given to EU rights may or may not persist beyond that date. It would be best to register for the EU settlement scheme before then, whether using your EU passport or ID card, to ensure that you do not end up as a test case.