Have you considered things like cooking pork or beef neck bones in a slow cooker. With spices, the meat is both tender and quite tasty. Also, buy whole chicken. I get the cheaper cuts of roasts and pork, on sale, if I can. If you're in the US you should have a Walmart around.

If not, look for sales. I do. I search for them and buy when I find things cheaper. Fresh vegetables in this coming season, should also be cheaper. It shouldn't be impossible to say on Atkins. I hope you can do it.

Good Luck!