Does a GFCI-protected bath light/fan unit need separate neutrals?
Generally speaking, you don't want to have a fan/light - or more specifically, a light - on GFCI because if the GFCI trips due to something else on the circuit then you are in the dark. As I understand it (I am not an electrician, but I have seen other questions on this topic and I heard this from my own electrician years ago when he installed heat/fan/light for me in a bathroom), if the fixture is not above a tub or shower then it does not need GFCI. However, sometimes that is either necessary (no other reasonable location) or desired, and in such cases, yes you would connect it to the LOAD side of a GFCI receptacle.

The key is that all hots must go to the same place and all neutrals must go to the same place. Plus all hots and neutrals for any given use must be together in the same cable. In the case of a combination fan/light, that means one of two things:


Two separate pairs of hot neutral, each pair must be together and both pairs connect to the same LOAD side of the same GFCI receptacle.
Three wires - 2 hot, 1 neutral, all must be together and all connect to the same LOAD side of the same GFCI receptacle. This is your situation as described.


Of course, there will be switches in between. If you use 2 separate switches (so you can have fan or light or both) then your setup will actually be:


LOAD hot -> pigtail -> both switch hots (This could also be without a pigtail if the switch is a double switch with a common screw to use for hot)
LOAD neutral -> neutral on switch if you need it via a pigtail -> neutral (white) from fan/light
Switched Hot 1 -> hot 1 (typically black) from fan/light
Switched Hot 2 -> hot 2 (typically red) from fan/light


If your fan/light actually has two separate neutral connections, connect them both to the white wire of your 12/3 using a pigtail.