24" stud centers? Ok, well that is odd, at least around here in CA. But don't stress about it. Your idea of using solid oak is better than plywood, go for it. I would use 2 pieces of oak (not any hardwood will do. maple is super strong too.) each piece being long enough 24" and about 3" wide assuming standard 3/4 inch lumber. Sand it smooth and paint it the same color as the walls. Go ahead and lag bolt the oak to the studs and also use a couple hangers in the middle of the span for extra hold, or you could you liquid nails for a more permanent attachment.

Then bolt the mount to the 2 pieces of oak using large lag bolts.

Oh yeah and liquid nails is a good idea, but if you don't want it that permanent, use silicone.