you don't mention the type of game bird, but i do quite a bit of pheasant hunting and pellets in the meat is common. if you use steel shot, there is a real possibility of busting a tooth - not as much threat w/ lead.
shoot your pheasant, and step on its wings, pull its feet. put the winged part in your cooler, cut your drumsticks off what's in your hand if you want them, in the cooler. when you get back to something resembling a kitchen, get the wings off and skin out your meat, put it in cold water in the sink. after 20 or so minutes of soaking you will quickly be able to see where the pellets went thru the bird. explore those passages for pellets, and remove any you find. you are now very unlikely to find any pellets in your cooked bird.
as for ingesting 'em, i guess i don't get concerned about it. if i eat a pellet or two, i'm sure they come out the other end in a day or so...? anyway, i don't have any lead-related health issues.