Avoid "get-there-itis" at all costs!
As others and you are already aware the tires ARE a concern.
Hours of driving generate heat and that will exacerbate any weak areas of a 30 year old tire. If one should let go, particularly a rear one, the damage could be quite substantial. Those rear quarters and rockers are some of the most difficult pieces to find.
Resist the urge to nail it. Those tires cannot possibly be trusted to do anything other than make you look like a puppy on ice.
Why not arrange for new tires where you pick up the car, and ship the originals by UPS to your home being careful to explain the location of the TPMS ?
Most all other failures are not too threatening and can be dealt with by someone experienced with monitoring the performance of a classic car and willingness to pull over at the first hint of an issue.
A ZR-1 is new enough to make such a trip easily, however one that has lived such a dormant life style may have previously undetected woes.
Have an American Express card with you and understand the trip may conclude with a car transport as you would not want to leave it on route.
Enjoy the trip and the ZR-1!
Vaya con Dios!
Marty