At the ranges you're hunting I'd say go with whatever cartridge she's most comfortable making a flow up shot with if necessary. Going from a 77 grain tmk to an 88 grain tmk will gain you some penetration but I'm not sure by how much. Shot placement is the biggest key. 22 or 6 cal won't overcome a bad shot but either will work with good shot placement and a heavy for caliber bullet. Good luck and don't overthink it!