I looked up Carl Linnaeus and the botanical term "Linnaean," and the dictionary gives the pronunciation of his name as li-NEE-əs, not li-NAY-əs. So technically, in English, this name should be pronounced li-NEE-ə, not li-NAY-ə. (The latter pronunciation actually seems to be used in Swedish, though they spell it Linnéa with the accent mark.)
For this reason (pronunciation confusion), I would choose the Linnéa spelling over Linnaea, accent and all, even in the United States. It removes ambiguity that is present when not everyone is familiar with the rules of Latin vowels / pronunciation.
This is not the English form. Linnea is more often used. This is the form taken directly from the botanical name for the twinflower, Linnaea borealis. Linnaea is the feminization of Carl Linnaeus, the "father of botany".