Welcome to the "evil parent" club, aka mothers of teens...

Joking aside, I hear you very well. His gf is clearly manipulative and is texting him things that she knows will keep him with her (possibly starting arguments simply because she can then "forgive" him, telling him she's leaving him just so she can have a change of heart...things that leave him in a perpetual state of gratitude toward her).

You can't win this. I speak from experience here. My son also fell for a manipulator who only cared about her own needs. I tried to show him what she was doing but of course I only ended up alienating him. So I chose to welcome her into our family, not because I thought I could change anything, but rather so that I could at least see my son occasionally.

There isn't much else you can do except wait for the inevitable break up.

You CAN stand firm with your rules when you speak to his mother.

Good luck. You are not failing at all.