That would be an interesting test to perform, and I would enjoy reading about the results. I hope you try it. Chrono the same load when everything aft of the muzzle is at, say, 50 F and again when at ambient temps of 90 F.
Some food for thought and of course, discussion.
My prediction would be as follows. When the firing pin hits the primer of a 50 F load in a 50 F chamber and barrel, but on a 90 F day, the quenching effect of the barrel on the propellant gases will exhibit same cooling as you would see on a 50 F day. Once the bullet emerges from the barrel, it'll fly faster and flatter compared to a 50 F day simply because the air is thinner. Muzzle velocity should be same, but as you note the ballistics will differ.
With the barrel and chamber up to ambient 90 F temps, your velocity should be higher as there is less quenching effect on the hot gases. Velocity should be higher, as would pressure.
Some say this is due to the powder being heated to a higher ambient temp, but I've yet to read a definitive test supporting this. In fact, I've read the opposite. A purposely chilled round being discharged in a hot barrel will produce a velocity and pressure almost identical to a ambient temp round in the same hot barrel, or so I've read.
Powder manufacturers go to great lengths to ensure their propellant burns at a predicable rate across the temperature ranges expected. We see those burn rates published on comparative tables, but the other side of burn rate is the volume of gas produced as the powder burns. And it is the volume being effected by the barrel quench, and therefor the resulting pressure. A slower burning powder producing more volume of gas will produce a higher velocity (if the burn front keeps up with the accelerating bullet) with less pressure than a faster burning powder producing less volume of gas.
As I wrote, food for thought.... I've never read of a test describing barrel quench effect on gas volume production yet the science of physics should predict it. In my mind it is why the mass of a heavier barrel enables long strings of shots to fly more consistent than a light weight "pencil" barrel. It's not because it bends or deforms less, it's because the barrel time of the bullets, shot to shot is likely more similar because gas volume and pressure is more similar.
Anyway, that's my pet theory.