[RPG] Must I pick the higher die when I roll with advantage

No, a Crit Fail doesn't automatically overrule Advantage.

The Advantage and Disadvantage section of the PHB clearly states:

Use the higher of the two rolls if you have advantage... (pg.173)

Nothing in there says anything that would suggest rolling a 1 would cancel out Advantage so you would still take the higher of the two rolls.

The Rolling 1 or 20 section states:

If the d20 roll for an attack is a 1... (PHB pg.194)

So another way to look at it is that the higher of the two rolls becomes the "new" d20 roll for the attack and, so long as you haven't rolled two 1s, doesn't meet the requirements for a Critical Fail.

This is also backed by Mike Mearls (one of the lead designers, though not rules arbiter like Jeremy Crawford, for DnD) on SageAdvice:

Jerry Hunter
@huntinglabrats
@mikemearls A player had advantage on an attack and rolled the first die a 15 and the second a 20, does he get a crit on the attack?

Mike Mearls
@huntinglabrats Yes - the player uses the better of the two results

This is the sort of thing that can be fun, but it's also possible that things will become frustrating. In particular, having meaningful objectives that are thwarted again and again through no fault of your own can get very frustrating.

My suggestion would be that if you're going to do this, don't have it stand in the way of the PCs doing whatever it is they came to do. If your objective is to kill the bugbears, and you simply can't kill the bugbears no matter how hard you try, that gets frustrating. If your objective is to get into the fort (that the bugbears are guarding) or assassinate some other guy (that gets thrown at you by his bugbear bodyguards to cover their own retreat) or to track down information (including clues that the bugbears leave behind) then that's much less likely to be frustrating. There's also a lot of easy ways to play that sort of thing for laughs.