Could you provide a source to your claim that heter mesh rabbonim is used in cases when a man is refusing to give a get?
In my (admittedly limited) experience, I’ve only come across it used in the opposite way.
It’s a mechanism to prevent a man from becoming an aguna when his wife can’t or won’t accept a get.
Cases where the wife went crazy, is in a coma, or just refuses to receive a get (based on another Cherem dRabbeinu gershon not to divorce a wife against her will).
I furthermore had heard that in these cases, the man gives a get to his wife via beis din; and as soon as she is able or willing to accept it, it becomes valid and she’s divorced.
Your presentation of what HMR is used far seems extremely different than what I’ve encountered, and I’d be interested to see some evidence of your example.