Hi all - I took bits of all your responses and ended up making a tuning macro that brings the chamber temperature to my normal printing temperature first and then runs the autotune. When actually printing, I've adjusted my gcode to bring the chamber to temperature first (in serial) before heating the bed, rather than in parallel as I did previously. Overall print time has gone up a bit, but a single heatup in the morning and then leaving it like that all day isn't too bad.
Thank you!