You can create a CP - EUI compatibility mod, but this requires some time and dedication, and a whole lot of cross-checking. There are a couple of ways to get proper full compatibility: you can go over 3a) Vox Populi - EUI files, separate the lines required for CP and VP(CBP), and remove the latter. Alternatively, you can cross-check Community Patch's LUA files with the base games, find what CP adds, either replicate them in EUI files or use them as guidelines when separating CP / VP(CBP) changes in 3a. You can also compare Vox Populi noEUI's LUAs with CP' for that.
That said, CP's LUA folder should overwrite EUI's, and as EUI is modular, it should work without deleting the CP\LUA folder in theory. In practice, some files can be problematic due to various reasons, you can test them separately, and remove the problematic ones from CP\LUA. You may miss out some features or information, but it would be playable. So install CP through the installer, and separately download&install EUI. Legacy EUI could be safer but the newest version could also work.
I remember making a CP, C4DF, EUI compatibility mod waaaay back, don't know if I can still find it though. edit: apparently you didn't need to do anything to get CP working with EUI in the past, don't see why that'd change now.