Fencing Grace states that you only need to be using a rapier of your size, one-handed, in order to add Dex-to-damage as the only stipulation. Though the name explicitly states "Fencing". Would it go against the rules-as-intended to use Weapon Versatility with it?
Yes, you can.
Fencing Grace is very specific, it only works with rapiers, and with any other weapons that state that they work like rapiers, which don't think that exists yet. But weapon Versatility does not care about the type of the weapon at all, it will simply allow you to change its damage type.
You could say that you are using wide swings when applying slashing damage, or hitting with the guard or pommel for bludgeoning damage, both are valid fencing techniques.