Please don't be put off flying solo in Las Vegas. As a newly minted PPL you have to start accumulating solo hours somewhere and there is nowhere better to do that than flying 1 or 2 Hr slots out of Las Vegas. The west end of the Canyon is easily within reach on a 2 hr trip and flight is permitted at lower levels than at the popular South Rim. There are numerous other places to go on all points of the compass.
The weather in November can be very suitable for developing cross country skills and the beast of density altitude is mostly spent by then. Weather is better early in the month than later when the snow often sets in. Storm systems with high winds are predicted days in advance, but can last for a week when they arrive. Be flexible.
There are several FBO's at North Las Vegas and the best plan is just to go round and talk to them. They won't be especially busy in November, but avoid Thanksgiving. You will need a rental car - the shuttle to the strip only goes one way - when it's going at all. You could stay at the Texas - about a mile from VGT on Rancho - but I'm not sure I'd walk up there! There are plenty of cabs but VGT to the strip is $20 or more.
Of course you will need a reciprocal licence as well - all that's been covered to death here. I have no idea about FBO's experience requirements but I'd expect an extended checkout - something like the 3 Hours required before getting a sign off for the US private license practical test. After that, even if you then only get to fly 1Hr solo in the local area, you've done it, and the whole US is open to you. There's always next year.
VGT is under the northern edge of Bravo airspace and you can actually route NW to the practice area and Death Valley without entering the 'B' at all. But more importantly, controlled airspace is not the scary stuff we have here. There are well rehearsed entry/exit proceedures through the 'B' to the Lake (Mead) and the Canyon. Controllers may well make you wait for entry, but they won't refuse it as you might fear at home.
Any solo checkout you get will cover these proceedures and from VGT a checkout is an experience in itself, usually routing over the lake to Boulder Dam to familiarise you with the VFR route.
Grand Canyon airport GCN is somewhat ambitious for a new PPL at 2-3 hours away and 7000' elevation, sensibly requiring an overnight and careful planning, so may be a better bet when you have a little more experience. You can always drive there and be able to explore without having to worry about getting the plane back to it's owner. Even then, it's an 8Hr drive from Vegas and one year I got snowed up on the freeway in November driving there!
Finally and purely personal but I'd suggest avoiding passengers when first making these trips. You need to be able to make completely objective go/no go decisions and even more important be able to land out and stay overnight en route without worrying about someone else's priorities. If you miss your flight home to UK, so what? There's another one tomorrow!
Happy flying.