@ashsh is correct, the P71 chip in the Apex and flexSecure implants uses Java Card 3.0.5, which predates support for ED25519 (and other named curves). Brainpool should be supported for key lengths up to 640 bits (more than enough for brainpoolP256r1). I’ve never tried generating or importing any Brainpool keys though.
Here are some other posts that discuss it in more detail:
Here is the public security policy that NXP submitted to NIST for the P71D600 (a newer variant of the P71 that runs JCOP 4.5 instead of JCOP 4)
It covers all of the algorithms supported, and it explicitly mentions support for brainpoolP256r1 key generation. The section on ECKey Session (page 29/30), explicitly mentions the ability to import P-256 public keys, and the table on page 56/57 says “Entered during manufacturing/personalization or imported in secure challen specified by GP-AMD-I” in regards to APP-ECC-RT-PRIV_KA. Those capabilities are based on the SEMS Lite applet developed by NXP, though.
I also found the Security Target documentation for the P71 JCOP 4, and while it does generally seem to support importing keys, I didn’t see anything explicit about P-256/braintrust256