Agree with 2WA. If you're looking for serious off road riding, I would get a second lightweight bike for that.
If you're looking for modest off-pavement riding then mod the V650 or get a Tiger 800 or small BMW. Cost wise, the V650 modded is probably cheapest, but the Tiger 800 would be pretty close without needing mods. The BMWs are highest cost.
My V650 is really quite capable off road, much more than the operator is! OEM wheels with better tires, crash bars, radiator guard, axle sliders, factory hand guards. Not much added expense there really. I carry both an electric air pump and a small bicycle hand pump, plus a variety of tools.
You can disable the anti-skid by holding the front brake and doing a burn out of the rear wheel (on a slippery surface like grass or dirt). I am not of that camp that wants it disabled but you can do it if you desire. Apparently you can also unplug one of the wheel speed sensors to disable it, too, but I have not verified that myself.
Going to a larger front wheel diameter would change the road handling. I've never wished for a larger diameter front wheel off road yet, but my off road isn't motocross style. I do like the idea of spoked steel rims and wish Kawi made them standard, though I don't actually worry about damaging my rims with the kind of riding I do.
Skid plate? Yes that would have some value. They're not terribly expensive to add to the V.
I think it comes down to how aggressively you expect to be riding off-pavement, and how often. The V is an excellent compromise at a great price point. The Tiger 800 is a competitively priced off-the-shelf option if you're looking for some pretty serious and aggressive off-roading. The BMW's are nice but you pay a lot for them. Their 650 is a 1-cylinder motor, btw, so probably pretty vibey.
Another option? The Versys 300.