Often when I am running in relay it is precisely because my automated HP/MP restore isn't working properly and I have to adventure a few turns to enable scripted adventuring to work again.
So you say that you generally automate turns and want your between battle actions - script/mood/HP restore/MP restore - to execute there, but when you adventure in the relay browser, it's because you can't automate and want to adventure "a few turns" there before going back to automation.
Very good. Go to Preferences/Relay Browser and uncheck the four options which tell KoLmafia to do the between battle actions when you are doing manual adventuring.
Run betweenBattleScript before manual adventures
Run moods before manual adventures
Maintain health before manual adventures
Maintain mana before manual adventures
You do that, then you can automate as usual, KoLmafia will do moods and recovery, and if they fail, you can go to the Relay Browser and run without automatic moods and recovery and without the attendant failure screen, and when you are ready, resume automation.
Now, if you ever spend an extended time in the Relay Browser and WANT KoLmafia to maintain your HP and so on, you'll want to re-check those boxes, but that is not the use-case you describe.