Troubleshooting this is going to be difficult due to the way the storage tiers are arranged. Each app stores various caches, downloaded assets and items that are stored to the filesystem and iOS sends various storage pressure messages to apps so that they can clean up.
In your case, the apps aren't releasing enough storage and the OS is going in and cleaning house.
If you have a Mac, you can connect the device over USB and view the log files. Ask how to do that and link here and I'll provide that as an answer.
Assuming you only want to use iOS - I would do several things:
- Watch the storage data reported in the Settings app - perhaps saving screen images to document the state over time.
- Sign out of iCloud - that can put a lot of storage out of "view" of the settings app. Mail downloads also use space that isn't "accounted for" so you could sign out of all Mail accounts temporarily while you try to isolate what program/programs are storage hungry.
- Try and keep track of which apps you run. Turning off background refresh is a good start to controlling which apps are run, but Apple Watch, extensions (say when dropbox appears in another app) mean that you can get code running from an app even when you don't launch it explicitly from the springboard/siri/spotlight.
Sadly, your "reinstalling apps" and watching them one by one is the best way to sort this out if you don't want to enroll a computer to measure things and watch the log file for messages that might give you a clue to storage space warnings.