Express interest to join mentor's group with whom a fellowship application got rejected
Go for option 1. How you contact him depends a lot on your relationship.
Generally, keep it simple. Inform him that your application was not successful and try to get to the next step, another application, or asking for other options.
This could be an option, but as said, the formulation depends a lot on your relationship and the PI's personality.

Hi [prof],
unfortunately, our application for xxx funding got rejected. I am still very interested in joining your group and working with you on xxx. Should we try to apply for grant xyz? Do you have any other suggestions?
Yours,
nxkryptor