Please excuse my ignorance but I'm not sure I understand. Who will not allow him to be adopted and will insist on euthanizing him? Who has the right to make that call anyhow? Having been an Air Force wife, I know that things are different on base. But can they just arbitrarily impose their will like that?

As for the feral cat, get a Have a Heart live trap, put some good smelling food in it. Others have covered the trap w/a towel. I've trapped a lot of ferals and the towel didn't seem to matter one way or the other, so it's your call. Then get him neutered. If you really can't keep him, then see if you can find a no kill shelter until he can be adopted.

Bless you both for caring for an animal who has no voice in the matter.