Jsolo has the right idea - makes some changes to the car and see how the behavior changes (or not). Swapping tires is pretty easy as long as the tires will permit (not uni-directional).
The first thing that came to my mind was some play in the rear suspension - tire(s) getting out of alignment when under heavy load.
Can you tell if it's the front or the rear pulling? If you can determine that, you will eliminate a lot of possibilities.
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