The tire's shape shouldn't change when unloaded as long as there is some air in the tire. But anything is worth a try.
My Spyder weighs 1500 pounds, with about 675(45%) on the front end. I run 22-24 pounds in the front.
I will add I just had the Vreds put on, and my local guy "balanced" the tires. I had to rebalance 3 out of the 4. Maybe try that, Martin. How old are the tires on the car? Tires do get hard as they age, and I replace them after 5 years. Sometimes my Audi tires would be imbalanced for the first ten minutes when cold.
I happen to have a Snap-on computer balancer from the 1980s, got it off Craigslist 10 years ago. I had to buy the centering adapters, and of course a wide5 adapter. And steel stick-ons. Now I just have him mount the tires, it's cheaper. The cool thing with the balancer, you hand spin it and it's pretty compact. It also runs on 12v DC, so could bring it to the track in case weights get thrown...
Martin, that diagram was funny!