You can google discernment counseling-- it is a pretty specific, time-bound process that is specifically designed towards making a decision one way or the other. Our counselor said when I talked to him on the phone about it prior that in our case he thought he would probably do some sessions separately with each of us, but we didn't end up doing that (and I think I've shared that he kept telling us we were veering out of DC and into MC generally and tried to steer back to the three basic choices-- status quo, move towards S/D, move towards R).
In my case, I'd researched it and picked this particular therapist more because he had Gottman training than DC training, before I found out BD#3 (the PA for 2 years part). My H was actually the one who brought up counseling after I started to lean out once I found out the full extent of his betrayal, and I suggested this guy because of his discernment experience. At that point, I was very ready to get out of limbo one way or another, and was fully researching the D option. (You'll recall I never had "we stay friends" on the table-- for me, it was either he ends the A, she is out of our lives completely, and we work on the M, or we D and will lead separate lives as much as humanly possible with two young kids. I had no middle ground there.)
My H was very, very ambivalent and having a really hard time making a decision, so the DC part fit well for him, and truly for me too at that point because I wasn't sure I really wanted to stay married by then. It was important to my H by this point that whatever we chose to do, we worked on together. I will say that for me, I ditched the "status quo" option in the first session and by the third session, maybe even the second, I knew my first choice was working on the M. The only option off the table for me completely was keeping the status quo. My H, though it took a lot of talking to get to this, would vastly have preferred to keep both me and the AP, so basically the status quo, though he acknowledged that the stress was really eating away at him. Also, since it takes two to keep the status quo, he had to accept that with it off the table for me, it was off the table for him too. The only option that can be unilaterally pursued is S/D.
DC feels like it is kind of like IC in a couples setting-- the counselor hears from both of you, but helps you each make an individual decision about those three options. I think if you have a good counselor, it can be helpful but you need to be 100% percent prepared for the S/D outcome. In some ways, it could be a "check the box/I tried" experience like half-hearted MC, which I feel was kind of our experience in MC the year prior, before I knew about the A but knew we were having problems. (As an aside, it still blows my mind that we went to MC for nine months or so, every single week, and he never spit out the fact that oh, I'm sleeping with someone else. He was DEFINITELY using the MC as a "check the box" situation and it went nowhere.)
In our situation I think it was helpful, I think the MC/DC is good with both of us and did a good job keeping us on track, pushing on both of us, our communication and assumptions, etc., especially when compared to our first MC. I don't know how much of that has to do with his training or not. And, things were truly at a total head and simply couldn't have gone on the way they were-- at least, I couldn't have done it. For my H, the stress of all of it (AP was absolutely pressuring him hard at this point, I believe, from things he said) was almost overwhelming and I do think the DC-- at least the process of it, combined with the timing of his trip to her city being a deadline for me-- helped end the limbo. Of course, it could have gone the other way.
In your case, I'm not sure I'd recommend it unless he says something about wanting to talk to someone together-- in that case, you could say there is this thing called discernment counseling that is specifically designed for people in our situation. But otherwise, I still would stick to your NC. I do agree it could come across as pursuit. Maybe have DC as something to suggest next time he says he is thinking about R. I could see it as a good way for him to show some level of commitment to figuring things out together with you-- committing to six weeks of DC-- so that he doesn't just jump in and then back out again and ruin your equilibrium. It could be a structured way to engage together and decide what you want to do.
I hope you are hanging in there... thinking of you!!
Me (46) H (42) M:14 T:18, D9 & D11 4/19 - 12/19: series of escalating BDs 9/20 - present: R and piecing