I don't understand why you need him to tell her abut the 1-hour coffee dates agreement BETWEEN YOU AND HIM. As long as he respects the agreement, what difference does it make if he tells her it's something the two of you decided? To me, it sounds as if you need her to know that you are in control and that he listens to you first - sort of like the way a cat or dog marks their territory so that others know it "belongs" to them. (And I'm not trying to offend you by creating a mental image of a dog urinating on a fire-hydrant). It does sound like you need her to know that this is a "rule", even if he has every intention of operating within that rule. Like, "Don't you even TRY to see each other more than this rule specifies." If he "follows the rule" without telling her the rule, that isn't "lying by omission", it's him handling his other relationship the way he feels is best.
From reading your post, it looks as though you are trying to micro-manage his other relationship. I think you are projecting a lot of what is not working in your own relationship onto his other one. BBy focusing your energies on the rules he must follow, you're not getting to the root of why you find these rules necessary in the first place. Is it so that he doesn't develop a stronger connection with the other person and decide to leave you? This CAN happen, even with the rules you have set up. You indicated that there is more to this than meets the eye when you said that you spend what little quality time you have together talking about her or fighting. Putting rules in place and worrying about whether it's "lying" or not if he lays them all out to her or just follows them seems to be just another way of finding something to argue about.
Loving Radiance described what I am trying to say in another thread. I'll just link you to it:
I would be hard-pressed to say that LR is "lying by omission" because she doesn't go over her "rules" regarding Maca with every new person she meets or has coffee with. She simply ACTS in a way that respects the limits of the rules.
This is already a contentious relationship that I'm working very hard to be ok with
So, what is REALLY bothering you about his other relationship? What have she/he/they done in the past to cause you to not trust them?