Many cuts work well, but a boneless roast from the hindquarter is best: top round, bottom round, eye of round, or sirloin tip (knuckle). Loins or even steaks work too, except they tend to fall apart while simmering.
You can prevent that by securing them with twine or butcher’s netting (after the curing, before the cooking). Or you could choose the “sous vide” method — which takes a bit longer but produces better texture and flavor, with less shrinkage.
“Eye of round” roasts, or loins from smaller deer are perfect to slice into medallions for hors d’oeuvres. A plate of miniature open face reubens on slices of rye “party loaf” will get rave reviews.
Options: slow simmer, or sous vide. Frankly, sous vide has wide application and is an easy technique to learn and use. Cooking corned meats might be the ultimate use of the sous vide technique.