Is this sufficient? I feel like I missed something sire I did not use the S union T property.
What you did is enough: $\displaystyle T\subset S\Longleftrightarrow S\cap T=T$ , no need at all to assume anything about the union, BUT we also have $\displaystyle T\subset S\Longleftrightarrow S\cup T=S$ , without any assumption on the intersection, so perhaps the question was to prove that T is a subset of S iff any of those two properties, separatedly, is true.
Tonio