The first thing I would try is to leave the chamber open, with a small fan if necessary to provide some ambient airflow. Don't cool the bed here, but try and keep the ambient temperature constant. This should give you a tune appropriate for door-open or first warmup. You'll then want to assess if you have any excess oscillation issues once everything is warm - if everything is tight enough (for me, within 1C or so would probably be fine, but it's your machine). You can try manually tweaking the parameters from here, or even try averaging 'open door' with 'closed door' parameters if performance isn't to spec.
Slightly more elegantly, but with a more involved integration, adding a prerun macro in the printing process to switch from 'warm-up' tune to 'steady-state' tune should be possible. You'd need to work out the cleanest way to integrate this into your slicing/printing workflow.
Then the kludgy option is to create a 'warm-up macro' for the first bed turn-on where temperature is ramped up slowly enough (e.g. 5C increments) that the heater fault doesn't occur.
An even kludgier option would be to reduce the max PWM of your heater. I'd wager you could get that bed up to temp with ~25-40% of 4400W. (Measuring your steady state wattage would give you a real number). This would obviously be slower to warm up, but might mitigate things like chamber temperature (which will now have time to equilibrate).