IF the instructions for the heater call for it to be on a dedicated circuit, it's not allowed to share it with other things (this is often the case on that sort of item and is called out in the installation instructions). Keep in mind, things like wired heaters need to allow for derating, so that 1Kw is effectively quite a bit more, and thus, that load is already pushing the limits.
It's not uncommon to use what is called a shared neutral to supply receptacles from a dual-pole breaker. Often, when this is done, they break the tab between the two sockets and feed the top and bottom with separate lines, this allows higher powered things to be installed (since it is two circuits). Because the phases are opposite, the neutral doesn't need to be twice the size since any current on one side through the neutral can't exceed the breaker's limit, and if there's an equal amount on the other side, the neutral has effectively zero current in it.
You can't run a separate neutral - it must be within the jacket of the wire you're using, so you'd need 12/3. And, assuming you want to keep the heater...run a new circuit for the rest of the project. Keep in mind, the bathroom will require a dedicated 20A circuit for the receptacle there - it can't go anywhere else and must stay in the bathroom. It also requires a GFCI protection, whether via a receptacle or a CB (doesn't matter which).