When used with EasierChests(https://www.curseforge.com/minecraft/mc-mods/easierchests) the tabs show properly, but clicking on any of them causes them all to disappear.

Yeah, any mod that does fancy stuff with inventory rendering is not going to display InventoryTabs' tabs. I'll look into getting the two mods to play nice

I figured out a method (released in 0.7.0) that should hopefully support all mod's custom rendering. EasierChests works but there is a little bit of overlapping (I'll address this later)