Lady Charlotte is definitely under her husband's thumb, but she's not quite as harsh on Clarissa. Yeah, she pushes the marriage to Mr. Solmes thing a little harder than she should, but there's a reason Clarissa would prefer to hash it out with her mom over any other member of the family. Even when Mrs. H. is mad at Clarissa, she still calls her "My dear girl—for so I must still call you, since dear you may be to me in every sense of the word" (41.1).
Lady Charlotte's major weakness is that she misunderstands her daughter's motivations. She thinks Clarissa's objections to Solmes can be overcome with bribery. Yes, Mrs. H. tries to bribe her daughter in submitting to the final will: "I hinted to you, you must remember, that patterns of the richest silks were sent for" (41.4). And that's not all. Mom not-so-subtly says that Clarissa will get way more if she'll just shut up and get hitched: "Mr. Solmes intends to present you with a set of jewels" (41.7).
You can't totally blame the woman, though. Lady Charlotte makes lemonade out of the marriage-lemons she's been dealt and advises Clarissa to do the same. It's not like the girl's got a lot of options, Mrs. H. seems to say, but at least she can do some major shopping if she marries Solmes.