How much difference will it make?

Disclaimer: I have never run drag radials. But, from what I have noticed on other's cars, drag radials do not respond significantly to pressure changes like bias slicks do, and those can run all the way down to 10- mid teens psi. Radials (someone correct me if I'm wrong) seem to like higher pressures.
Without re-engineering your whole car (which seems like a real nice setup, by the way- I would avoid changing too much), I would start by unhooking the front sway bar at the track as mentioned above, maybe loosen the t-bars a couple of turns, and get a set of dedicated wheels with slicks (personal preference, they're what I'm used to) with maybe 12-15 lbs. to start with. Just be aware that at those pressures you'll probably need to use bead screws.
Try your rear bar both ways- see what works for you.
It sounds like a real nice engine build, but it still shouldn't take a ton of work to get your 'teener to hook.
JMO.