Hello Rose,
we didn’t miss the reply, its just that we couldn’t do anything with it. The feedback felt like a standard answer since all the points you mentioned had already been tested and refuted. The point about changing the secondary menu was also ignored, beside the fact that you only tested 1 page at a time instead of the 3 / 10+ that we recommended.
TranslatePress stores the translations in a separate database table at string level, not in the theme mods as you claim. After all, translations simply work perfectly when the theme is changed.
Our developer screen-capped the entire topic and sent it to TranslatePress. Hopefully, they will know what to do with it.
For now, unfortunately, we are left with cutting out all elements that might pile on the incompatibility issues, hoping for the best.