Riding lawnmowers can be used and I've done it, more than once. They need to be geared UP though. However, the O.P. said the design goal is 10 - 12 person capacity. Mowers are pretty small for that. The ones I built were one or two passenger MVs.
Yes you can gear up your alternator to get full power from it at idle, but you then overspeed it if you ever throttle up and you won't get as much power as you will by adding a second alternator.
In most cases the donor vehicle will either have an airconditioner compressor or a place where one could have been if the car had been so equipped. It's usually ridiculously easy to mount another alternator there.
A super-high output Ambulance alternator is nice but a second regular one from a wrecking yard is a hell of a lot cheaper and again probably will make more power. I like the GM "CS130" alternators, easily and commonly found in late 80s - early 90s GM small to midsize vehicles. Those were specifically designed to produce a lot of current at idle speed.
I buy 'em for $10 - $15 each at the local Pull-A-Part yard. AT that price you can bring a few spares.
With two of those, at idle speed, running through a cheap inverter, I'm able to hold an 1100-watt electrical load without draining my batteries. At $15 bucks a piece, if more is needed, mount a few more! I've stopped carrying my onboard Honda generator, as much as I love those things.
I think you'll be OK with just about any "real" car tires on the playa. My current MV only has 185-14s on it, weighs about 5000 pounds (without passengers), and carries the 10 - 12 people you want to carry without failure. The tires may be "overloaded" for high-speed freeway use but crawling around on the playa is no problem.
Gearing down by significantly reducing the diameter of your wheels won't work because you wont have enough ground clearance. Nothing is impossible, you COULD get around that, but dragging the brakes a bit is easy and nearly free.
The playa surface conditions will determine whether brake-dragging is even needed. On soft-playa years, I've had to throttle up just to keep rolling. On hard-playa years, I did some brake dragging.
I've had carbureted V8 mutants that idled through crazy amounts of fuel, but the current fuel-injected 4-cylinder I've been using goes through about 20 gallons of gas all week, and that's driving the wheels off of it.
Too much is not enough, and when in doubt double it.