Hi Django,
The Nitrokey Pro can store 3 RSA keypairs. All keys use the same identity, but are used for the 3 different purposes you mentioned: authentication, encryption and signing. A similar answer is available in the FAQ (if you haven’t seen it yet).
If by “three independent RSA keys” you mean 3 different identities, it is not possible. Maybe you will find the answer in this ticket helpful.
You must be able to generate sub-keys from each keypair, and if not, I will confirm this and get back to you asap.