Ray, I’ll address your spare tire concern with what I know.
Short version:

Right now, before getting your GX, is go by a tire shop that has the same exact Cooper Discoverer RT you see in my pic. Use that as your gauge. Then stand whatever 275/65/R20 you’re considering next to it. That’s an easy test. No mounting required at this point.
Long version:

A couple initial caveats: 1) My GX is a ‘24, yours I guess will be a ‘25 or ‘26. So I cannot be absolutely positive that the provided spare tire space is/will be
exactly the same diameter as my ‘24. Maybe Toyota-Lexus slightly increased that space, volume without announcing it. 2) You cannot fully rely on stated manufacturer nominal specs, or, that all 275/65/R20 will be the same true diameter.
I currently have the Cooper 33.7-ish spare (stated to be 33.66) mounted, inflated and fully seated up into the spare tire space. It rests against all 3 or 4 touch points that guide it into place as you crank it upwards. It is fully seated vertically and laterally with no remaining space, none. I do not use a 5-tire rotation, so it’s okay to be a different tire. I only wanted an E rating, with nearly the same diameter as the mains.
That’s why I’m doubtful that a true 275/65/R20 would fit upwards all the way. However, an actual ‘shy’ 275/65/R20 non-E (significantly deflated, spray lube on touch points) might have a small chance of fitting.
Before I committed to my new tires path I went to my local DT store, and we mounted up the Cooper spare I thought would just barely fit. It did. It’s a very tall 275/70/R18 E, actual about 33.7 (vs. a nominal tire calc of 33.2). It just, just fits as I describe above.
Next, I ordered in my 4 main tires, the Toyo AT3 E, in size 285/70/R18. It is an actual ‘shy 34’, coming in at 33.7, same as my spare.
That’s how I did it, after looking at several websites - and then doing a hands on confirmation.
You can see the Cooper spare and Toyo main side by side in the photos before we mounted all the tires.
Best to you, you will love your GX!
-DJ