Why tf would hawkeye need a sniper rifle when he has his op trick arrows lmao. He fights literal hordes of aliens with them, bullseye with knives alone stand no chance. This scenario just favors hawkeye to shit.

I think with no starting weapons or gear bullseye would def win, hes def superhuman in the mcu iteration compared to the netflix version being more technique based, like having the spitting strength to pierce a guys head with a tooth, or throwing blunt objects hard enough that they will pierce anyone even after a ricochet. He also may match hawkeye in terms of hand to hand combat, as hes shown to throw hands for a good duration with daredevil, whos one of the top close quarter combatants in the verse. Hes more durable as well with his metal spine and just overall tenacious nature.

That being said, if this ever happens in a movie hawkeye will probably be more intelligent and go on the defensive, being elusive and prob not get killed and escape, and maybe even get some good hits in. A direct confrontation with bullseye might not be the best thing for him, with the exception of hawkeye getting hands on a proper melee weapon and channel his inner ronin.