Well, at first glance, you say "Oh my gosh, no! It should be I suggestd she GO to the library." But that would only be true if you had been making that suggestion to "her."
If, on the other hand, "she" was missing, and people were talking about where she might be, I can see this discription of the discussion: "Peter thought she was taking a nap, while Evelyn though she was at the pub. I suggested [that] she went to the library." (Even so, I might use "had gone.")
What do others think?