If any of your accounts ever receives a ban, then posting under any of them is considered "circumventing" the ban and grounds for removal.
As for damages, good luck. Under US law you have to actually prove damages, which are zero. I don't know if reddit has any legal presence in the UK or not, that's most likely the only chance you have of getting anywhere with them.
The biggest issue with reddit is that random mods in subs can basically kick you off the entire site by reporting posts as general reddit rule violations. I quit using the site a few years ago after a mod in a sub got mad at a post I made, and went and reported posts I had left in a FPS video game sub as "inciting or encouraging violence." I of course appealed but it was determined that the posts still violated the site policy.