Time for a heart to heart with your DH for you guys to decide what you are going to do next. I suspect it's going to be a little challenging to convince your DH they need to leave ASAP because he's dealing man to man, with the good cop. The other husband may indeed be a very nice guy, but at the end of the day he knows what his wife is up to & he lets it go on so he's complicit.
Particularly in light of the details you've been adding, I'd want them out, no renting, just out ASAP -- I don't mean toss them out, just tell them the time has come. The letter they requested from you could be a problem, could simply be a way of explaining their financial picture. I didn't understand the mortgage being under the table. I'm not familiar with that one.
Not to toot my horn, merely to emphasize what is what, these people are definitely grifters. They latch onto decent people & take whatever they can get. I suspect if you kicked enough rocks you'd find that there are a trail of decent people in their wake who have been taken advantage of & don't say anything about it because they feel empathy for the sob story that's been given & were trying to help them out.
Truly, I promise you that you need to get these people out of your house as quickly & cleanly as possible, with as little drama as possible. If I were you I'd be having a crisis of your own pop up any day now, necessitating getting your home in order for a prospective move. (Sometimes housing plans fall through unexpectedly you know
)
Good luck to you -- and be very watchful of damage, injuries, etc. as they exit. Would not be at all surprised to find they suddenly have an ailment, a chronic ailment caused by living in your basement, leaving you & your homeowner's policy up to bat. Disconnect as cleanly & pleasantly as you can.