Word to describe "concealing" positive emotions, in fear of being perceived as irrational

Words like seemly, proper, careful, knowing, canny, shrewd, astute, and decorous may fit. With slight rewording, phrases like with appropriate reserve, in reasoned measure, carefully proportioned, and not unseemly can be made to work. Of the above, canny ("careful, prudent, cautious" or "knowing, shrewd, astute") may serve best, because of its connotations of conscious, knowing insight and an eye to the main chance.

How about "guarded"? It means "noncommittal", "restrained".

The syntax you’ve committed yourself to makes it tricky. How about “with great, yet closeted, enthusiasm”. Or with “covert” for “closeted” (though I rather like “closeted” here).

With different syntax you could go for: “... yet does not display any (outward) sign of enthusiasm” or “suppresses any display/sign of enthusiasm”.

Slightly less transparently, you might play on the expression “to play one’s card close to one’s chest” (meaning, not to let on about what you are doing or have prepared): “he plays his enthusiasm close to his chest”.

Perhaps the phrase tempered the expression of his enthusiasm might work.