I'd roll with Underwood 9mm 147gr bonded jhp at 1175 fps, or similar.
Can't be any worse than 30 Carbine, 25-20, 45 patched ball, 22rimfire, or a host of many small rounds that have killed deer effectively since dirt was young.
It's all about knowing when to pass, understanding the limitations of your chosen weapon/ammo, and placing the bullet well if an opportunity presents itself.
This one fell to a single 5mm rimfire mag,