I am inclined to believe that Mrs. Dashwood did not keep her pique about Fanny Dashwood's insinuations to herself.

I think her wish to spend as little time as possible under the same roof with her was aired to Marianne. Marianne would have been even more indignant, and more eager for removal, for Elinor's sake.

Did you notice that Elinor did not want to leave, but buried her desires in favour of the family's well-being?

[...]it was not a plan which brought any charm to [Elinor's] fancy, though it was a removal from the vicinity of Norland beyond her wishes[...] (Ch. 4)