Pretty sure you can build a cutter that can stay inside a burning station indefinitely.
This is correct, you can. Ideal for collecting all the items and materials.
With a Grade 5 Low Emissions Power Plant and Grade 5 Clean Drives, both with thermal spread, the Python can sit inside a burning station for around 1 minute 50 seconds before needing a heat sink.
With the same engineering the Corvette, Cutter and Anaconda do not need heat sinks and can sit in a burning station indefinitely. Of all of them the Corvette has the best cooling and agility (you can add 10% heat back with drag drives and it makes no difference, heat gets to 65% and stays there forever) and the highest passenger capacity (212 if shieldless compared to Cutter at 204 and Anaconda at 202).
The problem then becomes being able to clear out the passenger board in 3-4 trips, as noted above. I normally clear out the passenger board then switch to collectors and cargo and take missions from the rescue ship for pods etc. while the passenger board is refreshing, then switch back. If there is another station burning nearby you can alternate between the two stations doing passengers only.
It's a bit of a nonsense situation as the concept of having to wait 15-20 minutes to rescue more people is internally illogical but the passenger boards are copied from regular stations where you would not normally run them out of missions.