I'm curious why you selected a PID control loop?
That seems to be the go-to control solution for people that don't understand the problem. "We'll fix it later by tuning the parameters."
You have a smart controller, and know the problem domain. Use that knowledge. Build an implicit thermal model. Better, a self-tuning adaptive one.
For instance, just after power is turned on, you can check the temperature. If it's about room temperature, you are in a normal warm-up phase. You need to put a specific amount of energy into the heater to bring it up to the target temperature. You can express this as joules or watt-seconds. Since the power is effectively constant, we can simplify it to seconds or internal ticks. Just turn the heater on full for that number of ticks. The controller might do an occasional temperature sample to check for faults, but the duty cycle should be essentially 100% for this phase.
If your thermal model is accurate, the heater sensor should be in a moderate over-temperature condition at the end of warm up. This is good. It's exactly what we want. The tip surface temperature will still be colder than target, and it's the surface temperature that is what we want to control. It's a key difference in model-based control rather than blindly implementing threshold feedback or PID.
Once at the target temperature, switch immediately to tracking mode. Use a previously estimated duty cycle to the heater, and watch the response curve. If the response curve is stable, use it to refine the duty cycle needed to maintain tip temperature and increase the confidence in the current thermal loss estimate. If it changes, switch to an currently-in-use mode. Immediately apply full power that would bring the unloaded surface temperature up a specific number of degrees. Then measure the actual temperature, comparing it against the expected value. Continue doing this, monitoring both the additional power and time of use. Use these parameters to develop a use profile. If the user is quickly soldering small joints, they should get matching modest bursts of power. If they are soldering heavy wires, they should get more energy even if the thermocouple response isn't fast. A microcontroller can trivially track state -- use that ability!