Electronic timer not required
They also make clockwork timers which are simpler to wire. The difference being your wrist powers the clockwork, whereas a neutral wire powers the electronic job.
Regardless, the timer will need to be placed on the LOAD side of the GFCI. That means you need to get hot and neutral correct.
I don't see a problem with the towel rack being outside the shower.
Any receptacle in a bathroom must be on a circuit that can be either of these two.
- A circuit that serves ONLY loads in this bathroom (receps and hardwired loads are fine).
- A circuit that serves ONLY bathroom receptacles in any number of bathrooms, but nothing else.
That rule is going to haunt us, if this is cord-and-plug connected.
Power for it
The towel rack needs to get power from some GFCI somewhere. If it is cord-and-plug connected (not hardwired), the above receptacle rules must be followed.
So the obvious choice is the GFCI receptacle that's right there. And only 100W isn't going to overload the circuit.
If the device is hardwired, then the restrictive bathroom receptacle rules go away. You can power it off any circuit where such a load isn't forbidden. (for example kitchen countertop receps). But a basement or outdoor recep circuit would be a fine choice provided it has GFCI protection.
If you use a circuit that doesn't have GFCI protection, you can fit a GFCI "deadfront", which is a GFCI recep with the actual receptacles blanked out. It's just for this kind of thing.
Power straight to the towel rack
The above rules still apply if you bring power straight to the towel rack from somewhere else. I recommend that source be GFCI protected, so all the wiring is GFCI protected, including the supply wires.
You can then have a "switch loop" going up to the twist-dial or electronic timer, and use /3 cable so neutral is provided. Code doesn't require it, but your electronic timer does.