It would have been a delicate undertaking but she was in a desperate situation. Fanny was making it obvious that she did not consider them welcome guests and one has to wonder if they had stayed on and on if she would have had the persuasive power to get her husband to tell them to go or if not that, to make it subtlety so miserable for them as to make it insupportable. Or Mrs. Dashwood may have reached her last straw with Fanny anyway in regards to Edward constantly being around and showing partiality to Elinor. If they'd ended up leaving without any help from John, and no offer of Barton Cottage, imo, they would have been at risk of sinking out of the gentry. I was thinking Mrs. Dashwood could have simply said something along the lines of, "You've been really good to allow us to stay but we don't want to impose on you forever. Henry had informed me of your generous offer to help us. I'd be so appreciative if you could inform me of what this entails so that I can begin to search for an independent residence for the girls." I don't see how even the incredibly selfish weasel of a John could have squirmed out of that, particularly when he cares about being seen as respectable. My guess is that he would have acted as if he had never planned on not doing anything for them and would have coughed up something. In the end, Fanny might have even considered it worth it just to keep Edward away from Elinor. If by some chance he'd actually came up with some excuse to get out of doing anything or simply offered a cheap way out by saying he would be willing to help them transfer their belongings when they found a place, Mrs. Dashwood could have simply said thank you and avoided a breach but at least it would have been out there and she would have tried.