English Translation
She is not always happy.
必ずしも appears to mean (not) always or (not) completely.
These are different concepts, I wonder if this sentence may mean that “she is not completely happy (about something)”.
She is not necessarily happy.
She is not always happy. = 彼女はいつも幸せとは言えない。
She is not completely happy. = 彼女は必ずしも幸せとは言えない。
必ずしも幸福ではない is understandable but not natural.
“Be happy” in this context in Japanese doesn’t mean “be satisfied with something”, but “to enjoy her life”. For example, she might be poor (i.e. not financially satisfied) but she can still enjoy her life with her lovely family.
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