I agree with Bill3 on this. You have two choices... either accept what your gf is doing or end the relationship. You can't change her nor can you control her. The thing that stands out for me here is that you & your gf have 5 years of history. I don't know how long she's been seeing this other guy. But it's apparently not long. If, after 5 years of history with you, she still wants to continue to see this other guy the writing is, it seems to me, on the wall so to speak.

I guess the good thing here, to the extent there is anything good, is that it sounds as though your gf is being pretty upfront with you about what she wants & what she intends to do. She's not trying to maintain her relationship with you while at the same time sneaking off with this other guy & then lying to cover her tracks... that sort of thing. At least it doesn't sound as though that's the case.

So I guess, from my perspective, if you don't want to simply end this relationship, perhaps the best you can do is to see if your gf will agree to a timeframe within which she will continue to see both of you. Then, at the end of the timeframe she will make a decision one way or the other either to become exclusive with you or continue to "play the field" indefinitely at which point you will have to decide whether or not you can accept a long-term open relationship. At least that way both of you know what the rules of the game are. I do agree with your therapist, you have no right to ask your gf to tell you if or when she has sex with this other man. You really have no right to any information regarding their relationship, in my opinion. The uncertainty is simply something you're going to have to accept if you're going to go forward with this awkward relationship.

You asked about nuances & pitfalls you should be wary of. Although from what you wrote it doesn't sound as though your gf is doing this, here are links to 4 articles, from PsychCentral's archives, on the subject of manipulation & "gaslighting". This may perhaps be one of the things you'll need to be cognizant of:

How to Spot Manipulation

How to Understand Gaslighting

Signs of Gaslighting and the Cost

Infidelity and “Gaslighting:” When Cheaters Flip the Script – Sex and Intimacy

My best wishes to you...