Private RSA key for HMAC key
The reason I am asking about the RSA private key is the HMAC key needs to be stored so that the HMAC can be validated by the server on future requests. An RSA private key is an easy to manage, persistent value. 


You seem to be under the misguided and mistaken belief that an RSA key is somehow easier to manage and persist than a symmetric key.


  I am wondering if there are any good reasons not to use an RSA Private Key for this purpose.


RSA private keys were not designed or intended for this purpose, and there seems to be no legitimate reason why you would actually want to do it. Use primitives for the purposes they were designed. Don't try to be clever. And don't invent your own crypto.