More searching yielded these plastic cleats.

For reference: TA Specialites models #38, #40 are meant for road use. #39 is "pista" for track use (deeper cleat, too hard to get out of)

The Exustar E-C101Al is a "pista" cleat (likely too deep).

Ebay seems to have a bunch of authentic, vintage TA Specialites cleats.