Confused About Tire Pressure in Winter Before a Long Trip

Hi everyone,
I’m relocating soon and will be driving about 350 miles a day on the highway. The total distance would be around 750 miles. The current temperature where I live has dropped significantly, down to 11°F (on average it's 18F/33F), but where I’m relocating, it’s warmer (lows around 20°F and average winter temps between 27–44°F).

My Mazda 3 is showing a low tire pressure warning at 30 PSI. When I drive around town (even at 30 mph for 10 minutes), the pressure goes up to 31–32 PSI, but I still have the warning. With my long highway trip ahead, I’m not sure if I should fill the tires to the recommended pressure (listed on the door placard) or leave it as is to avoid overfilling as the tires heat up on the highway.

Would you recommend filling the tires to the recommended PSI before the trip, even in this cold weather? Or is it better to wait since the place I’m relocating to is significantly warmer?

Also, I’m considering splitting the trip into shorter drives of around 250 miles per day. Any advice would be greatly appreciated. Thanks!