Lydia still thinks the best of Wickham when she has clear evidence of his wrongdoing, I think there are very low odds of Elizabeth persuading her to think any less of him, especially if it's just based on the say-so of Darcy.

Elizabeth might try harder if she had more information from Catherine thanks to their correspondence. But as I recall (It's been a while) she was already plenty convinced of Wickham's dangerousness by that point in canon, she just had no reason to think convincing anyone else was worth the hassle, since he was leaving/had left town. I don't see anything in your premise that would change that.

As for hiding her correspondence with Catherine, I can see a number of plausible approaches, but would personally lean towards "don't actively hide it and unashamedly admit it if confronted, but don't cause trouble by pointing it out if noone notices".

Hmm maybe she'd start carefully trying to steer people towards a better opinion of Catherine without bringing up the correspondence, both out of a sense of fairness and not wanting things to blow up as badly if the correspondence/their burgeoning friendship is discovered.