I agree that you'll need to upsize your tires for off-road use like that. If you can fit 2" tires on your frame, that'd make a huge difference. You'll be able to run them at a lower pressure and you'll gain a ton of traction and control off-road.
If you find one that has a center tread ridge, you may even find it not unpleasant for road riding. I have some Vittoria Mezcals on my MTB, but I pump them up to 60 psi and ride them on the road sometimes, and they roll out nicely. They're designed as a hardpack tire, so they're not mud tires, but they do stick pretty well on most off-road surfaces.