Best approach for Dropbox-API-Select-Admin for files endpoints using Team credentials for Business

Dropbox doesn't have an official best practice/recommendation for this, nor a "Default" option like you described, but I'll pass this along as a feature request. I can't promise if or when that might be implemented though.

So, you can do whatever works best for your app. I suggest something like option 2, but instead of rechecking on a schedule, just catch the relevant error if/when the admin ID is no longer valid and then get a current admin ID and retry the call.

Also, there is the /2/team/token/get_authenticated_admin endpoint to check which admin authorized a particular access token, so that may be a good value to start with.