A transparency plan would have to be very detailed and long, I think.
Not necessarily. A GPS on his car (where you can log on via internet and see where he is), a new cellphone # (with detailed billing, which comes to you), and a keylogger on his computer. Those three things would be about 90% "fireproof."
And (the more likely scenario) -- if he BALKS at any or all of them -- you can be 100% sure that he's still having contact with her.
That all being said, if he was SERIOUS about moving her into your home, I'd be gone-gone-GONE, and I'd throw everything I had at him, and then some, legally.