ありがた迷惑 is not two separate words. It is one word, a compound noun. When you make a compound noun like this in Japanese, you only use the stem of the adjective. The stem of ありがたい is ありがた, so this gets added to 迷惑 and you end up with ありがた迷惑.
Here are some similar examples of "adjective stem + noun" compound nouns, and the equivalent "adjective, noun" two-word constructions:
- 黒髪 (くろかみ) vs 黒い髪 (くろいかみ)
- 細身 (ほそみ) vs 細い身 (ほそいみ)
- 嬉し涙 (うれしなみだ) vs 嬉しい涙 (うれしなみだ)
- 早口 (はやくち) vs 早い口 (はやいくち)
In each of these cases, the two-word noun-phrase form (on the right) is a syntactically correct construction. But because the compound noun (on the left) has been lexicalized (become a word in its own right), it will usually have taken on some meaning or nuance of its own, diverging from the two-word construction to a greater or lesser degree. (And of course the fact that one is a noun and the other is a multi-word noun phrase can in itself have implications for how they interact with other words in a phrase or sentence.) So the two constructions are not interchangeable, even though they are obviously related.
All this applies to other types of compound nouns, too, e.g. 行き先 vs 行く先 etc.
In the case of ありがたい迷惑, I would hypothesize that ありがた迷惑 is so strongly lexicalized that in the majority of cases, unless it was clear that you were using the form ありがたい迷惑 intentionally for effect, people would simply assume that you meant to say ありがた迷惑 and misspoke.