Actually @jmhundley, I personally think your original solution -- a "lap cover" if you will -- is a much safer option. Seems like every time I get around a group which favors straps over good leg and dulcimer placement, or a cover, I see someone stand up without unstrapping, and the instrument hits the seat or ground. Can't say I've ever seen an "unstrapped" player stand and drop their instrument. I firmly believe that a "proper position" is more important that any artificial solution.
The knees needs to be spread more than shoulder width apart; the first fret needs to be over the player's left knee (assuming right handed) while the bridge (more or less) needs to be tucked into the right hip, with the dulcimer diagonally across your lap. You do not want your knees any closer together, and you don't want the instrument setting at right angles to your body, with more dulcimer hanging outside the knees than between them. If you habitually wear slick polyester slacks, then a shelf-liner is useful.