Guest List - is it rude not to invite a family member?

If I'm reading your post correctly, it sounds like you realize that you can't invite part of her family unit (her husband and kids) without her. That would come off as rude and petty.
As for not inviting her and her family at all, it's up to you. When she figures out that she was the only cousin who wasn't invited, she will be pretty upset. Especially if she's the type of person who always has drama. Even if she doesn't regularly speak to those who would be invited, these things have away of coming to light. The rest of the family may very well see you and/or FH as the bad guys for not inviting her.
The real question is what would be worse for you: having her at your wedding or the fallout that will result from excluding her?

If you’re not comfortable inviting her, don’t. We aren’t inviting a ton of his family that we both get along with because we need our numbers low for budget reasons. I’ll feel horrible but I’m okay with explaining why. Just say your budget doesn’t allow a lot of cousins and such.

It's your day and your memories. Personally I wouldn't want the drama that could come from not inviting her. Depending on how many guest you are having you might not even have to see her.

I would say it’s up to your FS.

We had a family member/couple we really do not like at our wedding. The only reason they were invited was because it would have hurt my grandfather’s feelings if she wasn’t. To me, that was more important. However, my brother got married eight years later. Grandpa said he wasn’t coming (health, hearing issues) so my brother didn’t invite said family member. In the end, grandpa came. I don’t know if he was upset about the lack of invitation for her or not. I know the family member was, but oh well.

It's your day so if you don't want to risk the drama I would say it's totally fine to not invite them. I'm not inviting my 3 uncles and their families on my moms side because I don't want to deal with the BS, there's been tons of family drama before I got engaged and my mom and aunt even asked me to not invite them. So we are just saying that we are having a small wedding and just having my God parents. But at the end of the day do what will cause you less stress. Good luck!

Not rude at all, it's your day and you definitely have the right to not want unnecessary drama when you guys should be enjoying your day.