While I can understand your frustration with the dealer concerned for not following through with items that may or may not have been agreed on, I am not sure how a Carfax report is going to help your case?
You state that you asked the dealer to provide a 8/80k mile warranty, Carfax won't help with that, the new tires you agreed on, Carfax won't help with that either.
All Carfax will allow you to find out is if it has been in an accident, that has been reported, (not all accidents are reported), it will show any recorded maintenance on the vehicle and previous owner history.
You also talk about wanting to take the car back in 90 days if you are not happy... this is between you and the dealership, not Carfax.
However, if you choose to go this route, and you still want a Carfax report, it would be prudent to buy the $39.99 version that not only will tell you about the Jeep you have in your hand at the moment, but also a Carfax on any other potential vehicles you look at to replace the one you have. That way, you are in the know of what you are buying.
Carfax doesn't protect everything. I had a Carfax report on my XK before I bought it, it was clean as a whistle, but upon getting the vehicle home, I could see that it had been painted on the rear bumper, a good job, but I could notice it, after I cleaned it. The new paintwork wasn't reported to Carfax.
I have had to have warranty work done on the XK too, but Carfax can't protect you from that either.
My advice, get the Carfax so you have the facts to hand, but bear in mind, from what you are stating above, with the tires, and the warranty promise, Carfax won't help you, that is between you and the dealer... not Carfax.