I think that list of foods to avoid seems very do-able. The basics of primal eating should be meats, fats, veggies and fruits. Lots of people don't eat nuts and dairy. Your diet will be close the diet that people that follow the autoimmune protocol from Robb Wolf.
For fat sources you can make ghee to eliminate any dairy from it, use coconut oil, olive oil, beef tallow, lard, bacon drippings. You have beef, pork, fish, shelfish, and chicken to choose from. There is a pretty unlimited number of options in that combo. It seems you have no limits at all on veggies and can eat most fruit.
Nuts and dairy are sad to give up - like you said they are the utmost in convenience foods. You just have to plan a bit more to make it work. I find having homemade soup in the fridge works great for a convenient food. I also follow Nomnom paleo's suggestion of keep packets of emergency protein in the freezer. I cook a whole chicken and roast one day then shred them and keep them frozen in large single serving bags. If I am in a rush I just grab one and heat it with ghee and frozen veggies for a quick meal. Coconut milk smoothies may be a good quick snack as well.
Good luck. And remember it is probably only for a while that these have to be left behind. You can do just about anything for three months