i think I would ask for a bigger bag? BTW i am not an expert so I do not know if this is worthy but heres my opinoin, ask for a bigger bag, and get the fish in separate bags..and go straight home from the lfs. 350 miles is a long trip but if you have consistant air temp in your vehicle. And you dont piddle foot around , those fish are able to live thru priority shipping, im sure you will transport them better than that.
Just my 2 cents, i think they will be OKAY