Check device authenticity by HASH
If an RSA signature is sizably truncated, as is the case here, it can not be verified with the public key. Hence here (necessarily: deterministic) RSA signature is used as a secret-key MAC, and could (and IMHO should) be replaced by HMAC.

That said, no, there is no risk of leak of the private key, or even the lesser risk of the ability to sign. If there was, that would demonstrably be a break of the RSA signature scheme used.

More generally, given access to a black box that performs the textbook RSA private key operation $x\mapsto x^d\bmod n$, there is no known way that $d$ (or an equivalent) leaks, unless the black box has a side-channel leak.