Establish Trust By Signing Random Seed
If you can recover the private key from a message and a signature (and the public key), then that signature method is broken, and broken quite bad.

We believe that RSA, with a decent signature padding method, is secure, and hence the specific failure mode you mentioned cannot happen.

There are two obvious ways to try to recover the private key for RSA; we believe that both are infeasible.

In the first one, we have the padded value of M (note: RSASSA-PKCS1_V1_5 has deterministic padding, so we always know that Pad(M) looks like, and we have the signature S; we could try to find the value $d$ where $Pad(M)^d = S \pmod{N}$, d would be the private key.  However, this is a discrete log problem; we believe that is difficult

In the second one, we could try to factor the public modulus $N$; if we could, then we could compute $d = e^{-1} \pmod{lcm(p-1,q-1)}$, and that would be the private key.  However, we believe that factoring a number the size of N is difficult.

One thing that might be an issue is a man-in-the-middle; if there is something between the client and the server; the server sends M to the man-in-the-middle, who forwards it to the client; the client returns the signature to the man-in-the-middle, who forwards it to the server.  The server thinks it has authenticated the client, but if the man-in-the-middle can then send its own requests to server, it hasn't really.

It might be that the larger protocol has protections against this sort of thing; however that's what I would be worried about.