Pick your path and work down it.
Both end in the same place. Path B lets you verify before removing anything, so take it if the app IDs differ.
Replace the custom upload
The store listing is the same application as your upload, so there's nothing to run in parallel. Do this outside working hours if you can — there's a short window where the app is absent for users.
Remove the custom app package
In Manage apps, filter by Custom apps, select your Republic entry and delete it. This removes the uploaded package from your catalogue; it does not delete anything in Republic itself.
Add the store listing
Search the Microsoft apps catalogue for Republic, open the listing and set its status to Allowed.
Repoint your setup policies
Open each policy you noted earlier. Remove the old entry if it's still listed, then add the store app under Installed apps and Pinned apps. Save.
Verify with your pilot user
Ask them to sign out of Teams and back in, open Republic, and send it a message. They should see their existing history and score — not a fresh start.
Tell everyone else
Once the pilot checks out, let staff know Republic has moved to the Teams store and ask them to open it once so proactive nudges resume.