Well, I don't think I even implied that the dog could understand what you're thinking. I don't think dogs can think at all. What I'm claiming is that for anything that can think (and thus entertain the idea of thoughts that cannot be thought), there are no thoughts that cannot be thought. The difference between you and your dog isn't just one of raw processing power. It's easy to imagine a vastly more powerful processor than a human brain that is nevertheless incapable of thought (I think Yud.'s suggestion for an FAI is such a being, given that he's explicit that it would not rise to the level of being a mechanical person).

Once we agree that it's a point about precision, I would just say that this ground can always in principle be covered. Suppose the translation has gotten started, such that there is some set of thoughts at some level of precision that is translatable, call it A, and the terra incognito that remains, call it B. Given that the cognitive system you're trying to translate can itself translate between A and B (the aliens understand themselves perfectly), there should be nothing barring you from doing so as well.

You might need extremely complex formulations of the material in A to capture anything in B, but this is allowed: we need some complex sentence to capture what the Germans mean by 'schadenfreude', but it would be wrong to think that because we don't have a single term which corresponds exactly, that we cannot translate or understand the term to just the same precision the Germans do.