The desire to know who your kids are around is a good one, in my opinion. However, there are a few things at play here.
First, most LBSs aren't as concerned about their kids being around OP as much as they are concerned about there being an OP. Those LBSs are not being honest with themselves about that, but use their concern for their kids as an excuse to "keep tabs" on their STBX.
The other problem is that even if the STBX agrees to this, they are under no obligation to follow it. You could get them to sign their name in blood on a triplicate form......and they still can have OP over at the same time with the kids and there isn't a gosh darned thing the LBS can do about it.
So then you get to the question of whether or not you can trust your STBX to not have people that are harmful (legally speaking) to your kids well-being around said kids. If you cannot, then you should be documenting the problems and working with your lawyer to fight for full custody. After all, a person that would make your kids available to people that are harmful (legally) to your kids, is not fit for custody about supervised visits.
So what does harmful, legally mean? It means someone that is committing a crime in the presence of your children. People that are actively dealings drugs while your kids are there, or that are sex offenders not allowed to be around underage kids, etc. Unfortunately, for the LBS, the OP that is sleeping with your STBX does not meet the legal definition of harmful, and therefore you would look foolish to take your STBX to court for custody in that case. I've actually heard judges say, talking about a WAS parent that is having sex with multiple people, say to the LBS "Okay, so you've proven she likes sex. That doesn't mean that she is a bad mother."
So it boils down to whether or not you trust your STBXW to not bring legally harmful people into your kids' lives, and if you really do not then to fight for full custody.
M(53), W(54),D(19) M-23, T-25 Bomb Drop - Dec.23, 2017 Ring and Piecing since March 2018