First, Jack is absolutely correct, the practical load of the towel eater will be about 1 amp, and that is typical of towel heaters. 1 amp on a 15A or 20A circuit is, as you might guess, trivial. It is also well within the 1875W(15A) limit of that switch.
However, 110.3b says you must follow labeling and instructions. Those say that you must provision 10A of service to the towel heater. That is a surprising requirement, and UL would not require that lightly. It certainly puts a new wrinkle on things.
- if the provisioning for a hardwired load accounts for greater than 50% of total circuit ampacity, no receptacles at all are allowed. So there can't be any receptacles unless this is a 20A circuit.
- If it's a 20A circuit and there are any other hardwired loads, we're over 50% for hardwired loads, so no receptacles.
- If it's a 15A circuit, there is only 5A left to provision other hardwired loads.
- If it's a 20A circuit, it has only 10A left to provision other hardwired loads.
Now we collide with the bathroom receptacle rules. Quite likely your installation is grandfathered. However, grandfathering is not exemption, and the rule of grandfathering is you can't make the situation worse. For instance if you have outlets every 20' along every wall (12' now required), you can't remove one. So...
- This load cannot be put on a (20A) circuit with receptacles that serve more than one bathroom.
- It can be on a 20A receptacle circuit that serves only this bathroom, but again, no other hardwired loads.
- It can be on a 20A circuit that serves no receptacles in bathrooms.
- It can be on any 15A or 20A circuit that serves only hardwired loads, providing this doesn't overprovision the circuit.