The chainsaw benefits from the melee damage character skill, but doesn't benefit from the power fist. Can a chainsaw medic get the Peace Medic achievement (Heal 300 points of damage in a single mission without dealing any non-melee damage to the Swarm), provided they only use the chainsaw and melee attacks to kill aliens?
I know this is an old question, but just in case someone else was wondering the same thing-
You HAVE to equip both the medical gun and healing beacons to be able to heal 300 points of damage, since just one of the above doesn't have sufficient healing power to unlock this achievement. So in effect, this rules out equipping the chainsaw. You can, however, have power-fist in your accessory slot to use in case of emergencies. Also note that teammates standing in the AoE of the Healing Beacon when already at full health doesn't count as healing.
Personally, I'd recommend getting Faith over Bastille for this achievement because she heals faster and runs faster. So you could get near your teammates who can cover for you and not get in the way of friendly fire.