Sounds like a constant noise. I spike especially severely to such things. Here's my 2 cents; our goal should be to a) have the loudness hyperacusis resolved, and b) try to not spike it up permanently. If you can lower your exposure to the heater, then that may be a good thing. In my opinion, sleeping with ear plugs are good and can make the tinnitus less intrusive in the morning for quite a few.
I personally think the problem lies in that the heater goes all around the clock, right? And that would mean to use ear plugs, all around the clock, which I don't think is a good thing.
Alternatively, you could for example sleep with some foam ear plugs, and get custom molded (or not custom molded) ear plugs with interchangeable filters so you could merely lower the volume of the heater, if that could mitigate the spiking.
But do keep in mind that I'm still very much in the same position as you, so my advice may not be good advice.
Wish you well buddy,
Stacken