So this guy did Knife Only:

Including both Final Jack + Final Boss, ignoring the Albert of course but really it's no different to the Rocket Launcher in REmake or the Linear Launcher in CVX.

So yes, melee only is possible. However some sections seem to be flat out impossible Knife Only, mainly the pit you get knocked into in the Old House and Margarete throws Bees at you. In this instance you're forced to use a gun.

I can confirm that Marguerite falls down by herself into the pit after quite some time.

Yes, Marguerite falls down by herself after a while, this happened in my melee only run