Use hook grip, problem solved. It will hurt at first but if you wrap your wrists and use chalk it will be bearable if you are not a sissy. I still prefer underhand-overhand.
I agree that anyone except from powerlifters should use straps. Even some russian powerlifters use straps and they switch to alternate grip before competition. Example:
Every olympic lifter I know uses straps.
And if you are worried about biceps tear, don't forget about stretching. Alternate grip stretches your biceps with huge load. If you are adapted to that and keep using this kind of grip with progressive loads you are gonna be fine. If you don't forget about mobility and stretching you are gonna be fine. If you use straps, forget about stretching, build huge, massive biceps and out of a sudden switch to alternate grip tear is inevitable.