Jim, as always you've covered the topic - and highlighted the main issue - perfectly:
Apple does not give you the ability to specify which app will have the privilege of being the icon provider.The only way I know to tweak iOS using a specific app for icon previews is its reinstallation.
Pixey, there are two options, and I suggest to start with the first one to minimize the risk of data loss. Because as Jim stated, deleting an app will also remove its local data (but NOT its iCloud files):
(1.) There is an option to delete an app without erasing local data: Apple calls it "Offloading" and its main purpose is reclaiming storage allocated by seldom used apps: By offloading you can delete the app to have temporarily more storage, and reinstall it later without loosing any data. So I suggest to offload StitchBuddy HD and reinstall it ... hopefully restoring its icon previews. Offloading is an option in the iOS settings under "General" - "iPhone Storage" after selecting StitchBuddy HD.
(2.) If offloading and reinstalling StitchBuddy HD doesn't fix your issue, deleting the app is the next option, but ONLY if you're using StitchBuddy with iCloud synching, so your files are saved in the cloud and will reappear once you have reinstalled the app and activated synching. Nevertheless this is a sensible fix and ever user error might cause DATA LOSS. So please be careful, use it only if step #1 failed and be sure - as always - to have a proper backup of your data.
The every last ressort (and most reliable fix) is of course removing the conflicting app, e.g. AcuEdit ... but I'm pretty sure it doesn't meet your expectations, so I don't consider it a solution. Feel free to contact me via StitchBuddy's support mail, if you have further questions or need some assistence.