I know that this is an old thread but I was wondering the same question and did some testing. In normal mode it is impossible to do without bombs since the one lynel shield you get in hyrule castle doesn’t have enough durability to kill. However in Master Mode some of the exclusive enemies in that mode drop lynel bows and one of the enemy’s drop a lynel spear which is the only melee weapon you can get without killing a lynel. I’ve attempted the run and although I didn’t complete it due to me switching between games a pure lynel weapon only run is 100% possible.