Never ever road track a car on drag radials. First off, most tracks will fail your tech inspection if you show up with drag radials on the car. The risk involved is sudden total loss of control of the vehicle under heavy load. Drag radials have soft pliable sidewalls, which are great for straight-line traction, but very poor for lateral grip. At high loads on a race track, you run the risk of a loss of control event during hard cornering. This is how you see what it feels like to hit a tire barrier and how much money it costs to fix (and most tracks also bill you for damage to the track too).
In short, don't track the car on drag radials. You'll need a set of proper handling tires set likely between 35-40 psi. It depends on the tire compound, temp and road surface as to what setting offers the best traction. You'll have to experiment.
For the Hellcat, gas isn't much of a factor. I'd keep it above 1/2 tank due to the propensity of the Hellcat to float the fuel pumps under high G loadings (which occurs mostly under 1/4 tank). The Hellcats weigh over 4,800lb loaded, so weight distribution caused by an extra 1/4 tank of gas is going to be inconsequential. Your biggest issue is the drag radials and the must-have removal of those.
Driving tips for any new driver...
Prior to the track day, ensure your brake pads and rotors are GOOD, your brake fluid and coolant is nominal, and all suspension components are good. Shouldn't be an issue on a new-ish car. After the track day, do it again. Track days are hell on a car and its tires/braking system. Some folks will pay for a post track day maintenance depending on how hard they drove and for how long.
On track day, spend the first few laps very conservative. If available, do a ride-a-long with an experienced driver on that track to learn the course a bit. If that's not available, you have to do that part yourself. Learn the track, note where the apexes are, identify any "gotcha" zones... places like a long straight terminating into a sharp turn, where there's rises or humps in the road surface during the braking zone that can cause the car to hop up on the suspension (and lose brake force).
Remember that this is more than likely your street car, and not a race car. Have fun and drive well, but trying to keep up with other track focused cars on the course is a fool's errand and pushing your car past its limit rarely ends well for the street car on the track. If you start to get into uncomfortable situations, regardless what Dodge says, lift and recover. Listen to and learn what the car is telling you. In my experience, a Hellcat is one of the absolute worst track cars. They're fun, especially since driving them aggressively generally means you're drifting most of the course (or snow plowing through turns with the heavy oversteer these things have). While the sensation, sound, and feel are very much fun, the actual track performance of the car is relatively poor. There is no graceful corner carving in a Hellcat. There's pretty much lots of smoke and noise... and if you have a car club like mine, every time I tracked the 'Cat, they'd shout "FREEEE WILLY" every time I passed the finish line. It is in fact a land whale and feels like it on a road course. Especially noticeable when I'd bring the Lotus and swap cars mid-day.