<sarcasm> I feel exactly the same, but only when there is an A or S rank monster in the area -- how DARE they put these things in a leveling area !?! </sarcasm>

You can always note the "time remaining" for a FATE and go take a break. Or, even better, work on more than that one. single. quest. obscured by the FATE.

I haven't found any FATE out there (or A/S rank monster) that entirely removes the possibility to progress with side-quests.