But, wouldn't it be my responsibility to comply if I want to rebuild trust?

Reassurance and transparency are needed to help rebuild trust, I think. But you can't live attached to the hip. Have to go about building trust in a healthy viable way, not a paranoiac way, no?

Besides, chaperoning isn't going to rebuild trust. All it can do is assure that no hank panky is occurring while being chaperoned. That's not in the category of rebuilding trust.

Rather than have a chaperone on any excursion (how does that make you feel anyway?), it may be better to have your cell phone available and if H needs reassurance, he can call you anytime, anywhere. He can also be informed as to where you're going and when you'd be back, so he doesn't sit around fretting about where you are.

I know from your karate class experience that the above suggestions are probably not sufficient for your H, but I feel that's because he wants so much control that he's not satisfied with any solution that treats you as an equal and responsible adult.

We're talking about bars here, really "going out", not like a movie or something with a girlfriend. I didn't think he was being unreasonable, so I agreed.

The point is not the perceived degree of his "going out" but rather, to honor the commitment he himself imposed. If he doesn't honor it in the least way, then he can't be trusted with anything more serious, IMO. Again, not that I think this is a viable agreement, mind you.