It would be easier to help if you pick a state first. Colorado is the closest so that might be cheaper.
Mule deer is out if you don't want to go through the draw. Deer are all draw here.
If you don't want to draw an elk tag you'll have to get an OTC tag. That means tons of hunters doing what you want to do. The 4wheeler will make it worse, because anywhere you can use it will be full of hunters with 4wheelers too.
What weapon are you planning to use? Bow hunters have more opportunity.