Is it possible?
I don't see why not, though I'd prefer a run on the PS2/Xbox Capcom Classics Collection Volume 1.
A deathless run, if it is possible
It is. There are MARP score replays that are deathless IIRC.
seems like it would be slower than one that wasn't.
No. What you're probably thinking is that a deathless run would be very conservative (read: lack of planning), and a run with deaths would be going in fists flailing (again, lack of planning). Being conservative means you don't have enough knowledge in the game to be confidently aggressive, and going in fists flailing while dying means you don't have enough knowledge in the game to predict enemy patterns
A good beat 'em up speedrunner will memorize the enemy positions and the boss patterns so he can hard and strike often. He will also group enemies so that he doesn't get surrounded and he can hit multiple enemies at once. Is it easy? No, but it's definitely possible.
I haven't found any TAS playthroughs
That's because most people on TASvideos would say a run of this game would be too boring.