For the insulation, I have decided to use this one (Buna-N/PVC Foam Insulation Sheets). We will need 3 of the 1 inch plain backing ones (9349K4) to wrap a few layers around it. I'll try two layers for now, since the insulation seems to be doing quite well according to initial testing.
Updated some values, most importantly, the k-factor. I had assumed that it was in the correct units already, but when converting it to 0.046 W/(m^2*K) from 0.26 BTU/(h*ft^2*F), I got the following plot. The time constant is still a bit larger than what we'd expect, but it's much better with these adjustments.
For our next steps, I will measure the time constant of the heater without any insulation and then decide how many layers of it we will need. I'll need to construct and calibrate a temperature sensor like the ones I've made before and use it to record the values more accurately.