I'm not sure Lady Bertram's capacity to oppose Mrs Norris is new, or that any 'improvement' in her is due to Fanny's influence. There are two much earlier examples of Lady B resolutely opposing Mrs N's view:

Ch.3. (Fanny is to live with Mrs N.)
"I must live within my income, or I shall be miserable; and I own it would give me great satisfaction.......... to lay by a little at the end of the year.”
“I dare say you will. You always do, don’t you?”
“My object, Lady Bertram, is to be of use to those that come after me. It is for your children’s good that I wish to be richer. ........... I should be very glad to think I could leave a little trifle among them worth their having.”
“You are very good, but do not trouble yourself about them. They are sure of being well provided for. Sir Thomas will take care of that.”
“Why, you know, Sir Thomas’s means will be rather straitened if the Antigua estate is to make such poor returns.”
“Oh! that will soon be settled. Sir Thomas has been writing about it, I know.”

Ch.4.
“Fanny must have a horse,” was Edmund’s only reply. Mrs. Norris could not see it in the same light. Lady Bertram did: she entirely agreed with her son as to the necessity of it, and as to its being considered necessary by his father; she only pleaded against there being any hurry; she only wanted him to wait till Sir Thomas’s return,

In the first quote, we actually see Lady B demolish her sister's argument twice in succession! In both examples she is deferring to the male head of the household rather than arguing for her own preference. Her personal comfort and convenience are in no way threatened in either case - Fanny will still be able to walk up from the White house to tack on Lady B's patterns, and no one is suggesting that lady B should take up horse-riding.