My family has owned Chinese restaurants in the past so I think I can be a bit helpful!
Pretty much everything is cooked with vegetable oil that has been heavily oxidized by the cooking process (the wok gets extremely hot) which means that any sauce should be avoided. I also recommend asking that no MSG be used even if the menu says they don't. Some places still use it despite advertising otherwise. Even without adding MSG while cooking, many of the products they buy to make their sauces (i.e. plum sauce, oyster sauce) contain MSG. If you need a dish with sauce/gravy, I recommend only ordering dishes with white sauce. The brown sauces will contain products that use MSG and they also have vinegar as an ingredient which is alright except that the chef adds sugar to balance out the sourness. Any sauce that is a gravy has had corn starch added to thicken it. Same goes for the thicker soups. Authentic stuff won't have this issue.
For the good news: the meats and veggies are blanched in water or broth (probably more like broth at the end of the day) before stir-frying. You can request that whatever you ordered without the sauce. Typically they refer this as being "steamed" but that really isn't the case.
Most chinese restaurants don't have Tamari but I suppose you can bring your own. You can use soy sauce they provide to flavor the dish. Most restaurants will also have chicken broth available as it is part of the base for many sauces and soups - you can request a side of that to go with your meat and veggies. Chances are the broth contains MSG though.
Some places still use lard for frying but they won't be using it for cooking (they probably won't tell you though since "vegetable oil" is what everyone else wants). Still most of the fried stuff is breaded and wouldn't be primal anyways.
Other option: if the food is good, eat it. Just try to minimize the sugary sauces (which isn't hard with authentic chinese) and take an extra spoonful or two of omega-3's for the day.