I think this should be "Ithim AN dinnéar le mo theaghlach". Any Irish speaker have thoughts on this?
Though I'm not an actual gaeilgeoir I do have thoughts. Since you're not translating to a sentence with a definite article (the) the translated sentence wouldn't need a definite article (an). I eat the dinner = ithim an dinnéar. I eat dinner = ithim dinnéar. There are exceptions where an doesn't translate directly to a definite article in English. "Is maith liom an Nollag" meaning "I like Christmas". I think there's a reason, but I'm not sure what. Hope this helps at least a little bit.