Jake for an air-cooled ATV, I'd go with a narrow-spread oil as I suspect the heat and RPMs will shear down most VIIs.
So, Instead of 10W40, I'd use a 15W40 and you'll get the benefit of the extra anti-wear most 15W40 heavy duty oils have. The 20W50 should be fine as well ... as long as it is swapped out often. Since most of these machines have low-capacity sumps, I'd drain the oil every 20-30 hours of hard operation. Maybe more if you suspect sand contamination.
Testing a sample after 1 day's hard use might not be a bad idea to give you a better idea. Those machines ain't cheap!
I had been using Chevron Delo 400 but will slowly switch my vehicles (including a 1983 ATC 200E "Big Red") over to Schaeffer Supreme 7000 synthetic blend in 15W40.
I agree with Bror except that I don't know that it is the best use of synthetics unless your buedget is unlimited. A good Group II product like Delo with frequent changes would be my choice.
Another reason to go with 15w40 instead of 10w40 is a 27% higher viscosity under maximum Shear at 150C than a 10w40.
I don't think a 25% PAO blend like Schaeffer is overkill for these applications. In the states, it's a $1.75 per quart oil (the Delo 400) vs. a $3.00 per quart oil (Schaeffer Supreme 7000).
With all the expenses involved with 3/4-wheelin', you could change your oil after every hard day of ridin' and I can't say that the cost would be significant at the end of the season.
However, the Honda's I've owned/ridden are pretty abuse tolerant and either oil'd be fine, I'm sure.
Bror, We are using it in a Lake Racer with a Solid Lifter Flat Tappet cammed 454 Chevy with 350lbs of open spring pressure,the price is 2.25 per quart. Seems to do pretty well after two seasons,no flat cam lobes.