Sorry this may be a dumb question, but I've learned now it's better to ask more questions than not:

The season has already started, a FA I'm interested in his contract demands are as follows:

Type: Spring Training Invite
Years: 1
Total Amount: $0

Then in the contract offer it says $0 as well.

However, ST is already over, if I submit an offer to him will he get signed to the BL minimum? Or is he really willing to play for free at this point?

I believe he will simply sit in the "players needing assignment" area, and once you assign him to whatever team you want to sign him to, he will make the appropriate amount of money.

Offer him a MinL contract with ST invite.  If you keep him in the MinL, he'll cost between $38K-$55K.  ML promotion bill bump the salary to $327K.