Login failed errors while DB refresh in progress
One option might be to change the login for this user to have a different default database:

ALTER LOGIN [xxxx]
WITH DEFAULT_DATABASE = [tempdb];


This way at least the connections will succeed and the logging of errors will stop (the apps will likely have mixed results, mostly failures, with whatever they are trying to do of course).

And then you can change it back to the usual one after the refresh is done.