These are just my opinions, to be taken with a grain of salt. 35's vs 37's may only provide an additional inch of lift, but you can't always look at it as just an inch of practical difference. I am using extremes as an example, but if you picture two rocks which are 35" apart, a 35" tire will drop right down in between them. A 37 with it's 2 inch larger diameter, won't drop all the way down into that same 35 inch hole. It might only drop 80% of the way into a 35" hole, but that might be an effective net ground clearance difference of several inches measured at the axle centerline. (pulling numbers out of a hat) It goes without saying that a larger tire will always provide greater clearance. 35's perform fine on Gladiators and don't look out of place, but 37's do look better and are noticeably more capable in practice.
JL/U and JT Rubicon's can fit 35's with no additional lift. They can fit 37's with 2 inches of lift, but might require additional bump stop to keep them from rubbing when fully flexed up into the fenders. I ran 35 x12.50's on my stock 2020 JLU Rubicon with no lift with very, very minor rubbing on the wheel liners. Worth noting that Rubicons have "high clearance" fenders on the front, which sit around 2 inches or so higher than non Rubies.
The next consideration is breakover angle. The shorter wheelbase of a 2 door has the best breakover angle, providing the most clearance over obstacles such as mounds or when cresting hills. For the most part this isn't a major issue on the trails, it just places greater reliance on skid plates and you might have to winch in certain situations where a 2 or 4 door won't need to. Larger tires provide a slight advantage.
But the major difference between the 3 is in their turning radius. Turning Diameter (curb-to-curb): 34.5 ft. [2-door], 40.8 ft. [4-door], 44.7 ft [Gladiator]. There are A LOT of trails at AOAA and Rausch which are very tight. Not all of them, but many. In some places there will be trees 6" or less from your mirrors or fenders. A 2 door can usually cruise right through the tightest spots without stopping, while a 4 door might have to back up and reposition to make it. There will be trails where a Gladiator will have to be extremely cautious to avoid body damage. On the tightest trails, a Gladiator might even have to use their winch or stack rocks to keep from leaning on a tree, or make it around some of the tight bends. Not on greens, I'm talking blues.
As for the spare, you probably don't need one on the street with 35's or 37's because of the thick tread blocks typically found on good AT or MT tires. On the trail, you can just lay the spare in the Gladiator's bed and strap it down. There are a lot of sharp rocks and shale at AOAA and Rausch, my Kenda Klever MT/2's have two seasons of trail abuse on them and I've never had a flat. If a quality AT or MT tire in good condition is going to be damaged on a trail, it's my experience that it's much more likely to be in the form of a cut sidewall. Under inflation can put the sidewall at greater risk of trail damage, 15 psi or higher is safe imo.
Rocker protection is a must no matter what you're driving. The factory Rubicon sliders aren't bad, but frame mounted sliders are ideal. The factory Rubicon skids provide great protection, imo, but aftermarket skids are cheap insurance. The old saying is that you never really drive on the same trail twice, because the trails are constantly changing. People can drag rocks along, rocks can flip up, and move around. Sometimes the guy in front of you can make the trail harder for you, sometimes it's the weather.
On a side note, Bronco's have a system called turn assist. It uses the ABS pump to lock the inside rear tire as a pivot point, and it produces a much smaller effective turn radius. Wranglers do not have this function. However, there is an aftermarket device known as the Tazer JL which enables turn assist on JL/U's and JT's. You activate it by pressing one of the cruise control buttons when the cruise control is off, and the ABS system will lock the inside rear tire when the steering reaches full lock in either direction. I've never used it, never seen it used in person, and the Bronco guys were criticized by people saying that it can damage a trail. It does make a huge difference in the turn radius though, which is why I mentioned it.
Personally, I would not jump straight into PA's BOH trails with a Gladiator. I'd do a few greens, then work up to blues, and then once I'm confident in knowing what I'll encounter I'd attempt the BOH trials. Not trying to discourage you, just trying to paint an accurate picture of what you may encounter.
If you want to consult with a professional guide / instructor who spends 3 or 4+ days a week at Rausch and AOAA, you might consider Kyle from Off-Road Consulting. He's been wheeling at AOAA and Rausch since before they were off road parks and he has a business relationship with them. He does recreational and corporate, and he's the most experienced off-roader I've ever met. He'll give you 30 minutes on a zoom call for 25 bucks. If he tells you something, it because he's done it hundreds if not thousands of times.
