Google the words "no contact" "transparency" "infidelity" and you'll find lots of great resources. At a MINIMUM, I would think your plan would need to include him sending her a no-contact letter (content approved by you, and letter SENT/delivered by you), and quitting the bowling league. Maybe there's another league that the two of you can join together, if that's something he enjoys doing?
Transparency is all about the betrayed spouse, and what YOU need in order to feel safe in the damaged relationship. It does NOT need to be permanent (there will come a time when he doesn't need to do these things for you), but it DOES need to be as "bullet-proof" as possible.
I'm glad you're giving this another chance. Based on the "new Pearl," I do think you have a good chance of success, IF (and this is a big "if") your guy is willing to do the hard work necessary.
I would also suggest that you throw in one thing that YOU are willing to do, and I would make that one thing be a job. That seems to have been his #1 issue with you before, and I do think it's reasonable. It would help you to show some give-and-take.
