Don’t let him fool you. Everything he’s doing since being found out is an attempt to sweep this under the rug as fast as possible. His desire to ‘earn your trust back’ is simply a means to an end to he getting what she wants. Just like his cheating on you was about he getting what he wants, with complete disregard for you. He just wants what he wants and he’ll do anything he can to get it. It’s not remorse, it’s fear.
What he’s doing is flooding you with promises and apologies and emotional appeals in hopes that SOMETHING will stick. Especially these blanket offers of “I’ll do anything you need me to,” as if it’s your job to fix this, as if there MUST be some solution and you’re the unreasonable one if you can’t come up with it.
Life is too short to not get what you deserve. He cheated on you and you simply deserve better than that. Leave him and find that better.
Throw the whole man out at this point. Let him and that use yo be friend of yours have each other and the misery that will undoubtedly fallow them.