Basic question on interaction with HSM using PKCS11
No, the operations on the key itself are not standardized and not supported by the PKCS#11 standard. There are some key derivation mechanisms, but even the NIST specified ones have hardly any support (which is kind of ironic, as key management is of course one of the main reasons to buy an HSM in the first place).

These operations are pretty specific to BitCoin and are therefore not standardized. It is of course possible to extend the base functionality of PKCS#11 with custom mechanisms. But in general HSM's will not be created with BitCoin wallets in mind.

Probably easier to create a BitCoin specific hardware module interface with your own software support. Some HSM's do allow custom code to be loaded, so you do likely not have to create your own hardware for that. Smart cards can be programmed as well. Prepare for a steep learning curve though.