So I have finally got my graphics settings in a good place to where I consistently get good FPS both on the ground and in the air. Now I'm thinking about trying out Ultimate Realism Pack. If I install the texture pack but don't touch the rest of my settings, will I take a hit to my FPS?

It’s purely a texture and shader mod set. No FPS hit

Unless you go for the reshade tweak included. It has some GPU tweaks like the ambient lighting. For me (gtx 1080ti on 4k screen) my penalty is roughly 15% more gpu load. 

That might be the difference between smooth flight and jerky graphics when using pmdg/fslabs and detailed addon airports and lots of clouds.

But overall fps is not a big issue.