How much difference will it make?

Everyone's got good ideas, but...
If I'm reading him right, OP's car is still primarily his street car; and his main concern is the tires going up in smoke on his occasional track day, and how to improve the situation without re-engineering the whole thing. Maybe I'm wrong, sometimes I assume too much.
Six cylinder bars are great on the strip, but as mentioned can be a pain on the street- its nice to navigate speed bumps and parking lot entrances without jarring your fillings loose and/or crushing your headers... Thus loosening up the current bars a bit at the strip- not perfect, but is some help. As is disconnecting the anti-sway bars, the rear one try both ways and let the results be your guide.
I missed the part about stock-length rear shocks- yeah, those are probably topping out and hurting you there. Longer shocks needed, definitely. And make sure they're adjustable ON THE CAR, some need to be removed to adjust them. Pain in the arse for a street-duty car.
Nobody's mentioned an adjustable pinion snubber yet, but honestly, I've never needed one. Your results may vary, especially if you're on slicks.
Slicks might be a mask or a band-aid, but like a band-aid, they can serve their purpose. You're not "hitting them" hard enough with a relatively mild 'teener to make a huge difference, but they WILL hook instead of going up in your current cloud of smoke, which hurts your 60 ft. and et more.
And while you're finding your best air pressure in the slicks, throw a few extra psi in the front tires. Helps a bit with rolling resistance, and may even aid front end rise a miniscule amount.
Lots of time involved finding your ideal setup, which as you said can be a pain with the limited amount of runs you can get at a test'n tune session; but once you've got it figured out you should be able to roll in, swap tires and perform your tweeks in short order and be able to spend the rest of your evening getting in as many runs as you can and having fun.
Of course, if I'm reading you wrong and you want to re-do everything, then you can ignore me. :)