How cold will the room get?
As you said, a CHE can't be used in the tub. You can use it on the outside by keeping it a safe distance and using a thermostat to regulate it.
I believe there may be a small number who have used an RHP. You can't mount it directly to the lid since 1) it may bow and 2) the RHP may melt the lid. You would have to attach a piece of wood or something to the inside of the lid so you have something to mount to.
I've used a space heater with a fan in the past to raise the air temperature around some tubs. Again - the heater was regulated with a thermostat.
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