I had not noticed the difference in my books, the 2" planting depth is when there is more organic matter. You need enough that the growing potatoes can be "Earthed up" with compost. In your case they would be fine just nestled into the surface , hardly buried or not at all, with 5" of manure above. If you had enough card I would simply lay that on top of the decaying cardboard which is allowing ground elder through, then plant potatoes through holes cut in the card, trowel width, then pull any ground elder which appears through the planting holes. Once potatoes get going, the weeds will be diminishing and then mostly smothered, final clearing abw at harvest time.