Hmmm, I think the problem you described sometimes happens when you also use the Better Trade Screen mod, but you've said you're not using that with CQUI, so that can't be the problem.
The Better Leader Icon mod should be okay to use with CQUI.
The Better Tech Tree (UI) was updated on 11/29/2020 to improve compatibility with CQUI, so you'll want to make sure you're using the latest version of Better Tech Tree.
The Fantastical Map Script mod should be okay to use with CQUI.
The Real Strategy mod should be okay to use with CQUI, although the latest 2021 player feedback claims the mod is no longer working as intended (the AI is acting dumb, plus some UI display glitches).
I saw in the player feedback comments for CQUI that in mid December 2020, several people reported trade screen problems, such as: "Trade screen won't allow me to view what's being proposed for trade by either side. ... Sometimes gold, but not always".
From the subsequent answers from the CQUI mod author, the people having trade screen problems were using the Epic launcher versus the Steam launcher.
I also saw the following player comment: "For most Civ 6 mods, you can download them and put them in the mods folder to run them when using the Epic Games versions of the mods, however, for this mod it doesn't work. The problem is that it doesn't show up in the mods screen, which I'm assuming is because it's missing some info file for the game to read and actually recognize it as a mod."
Supposedly, you can get around this problem when using the Epic launcher by downloading, and manually putting the latest version of CQUI in your Mods folder, from the following Github:
https://github.com/civfanatics/CQUI_Community-Edition
Hope this info helps you.