You're finding out your endurance is not up to snuff. This happens to every guitarist I would imagine. I tire out faster alternate picking the low E string rather than the higher strings. What I would do is set your metronome to a comfortable speed and simply alternate pick the low strings (don't even worry about your fretting hand). Then slowly speed up to the tempo of the song until you can continue alternate picking for a couple minutes without getting tired. Don't hold your pick tighter than you have to. I should be able to walk up to you and gently pull the pick out of your fingers. Any tighter than that is too tight.

Good luck!

~Rusty