As mentioned above, when I have Apotheosis installed, I can't use the anvil at all. I can't even open one, regardless if it was placed before the mod was installed, a freshly made one, or a cheated in one after the mod was installed. It's only with this mod installed, as once I remove it the anvil works perfectly fine.

I don't have any of the incompatible mods, but if I need to make a new list of mods (as I had to completely reinstall the modpack I was using, and it's got a few different mods in it) I can do so.

This issue typically arises when you have a mod incompatibly modifying the anvil. Your game log should tell you the culprit.