There's of course Harman Direct Positive paper. Haven't tried it; it's high contrast, but it should work and I think some people have tried it in this application.
Yes, reversal processing works. With RC paper I usually got some degree of mottling due to the slightly uneven paper base, but YMMV as they say. When I experimented with this, I used a dichromate bleach, but a somewhat more sane potassium permanganate bleach should work, too. Strong peroxide and acid should work, too, but I've never tried this.
Contrast control will be the main challenge, but this is a matter of working out the parameters in terms of exposure and first developer chemistry. As in film processing, you'll likely need to add a silver halide solvent (thiosulfate, thiocyanate) to get rid of some of the excess density.
For an easier approach, consider simply shooting paper negatives and contact printing or photographing those.