Yes, get on with it. Life is short, people die with no notice. If your mutual friend can't handle it then that's his problem, but perhaps with the two of you in a relationship he will come to realise that there's nothing there for him any more. Whereas if your best friend remains single then your mutual friend will always hope that he can win her back. In other words ...
> he hasn't moved on, and she views that if we date now, it'd be like back-stabbing him
... maybe he'll never move on until she does.