If a responder to a strong 1NT bid is 5-4 in the majors, would it make better sense to use Stayman rather than a transfer bid?

At matchpoints, with only them vulnerable, I dealt and opened a strong (16 point) 1NT. Partner bid a 2 heart transfer (to spades) with the following: ♠ AT843 ♡ AQ87 ♢ Q5 ♣ Q7.

I duly made the forced two spade bid, and partner raised to four. We went down one despite our combined 30 high card points because I only had two spades.

Partner admitted, "I guess we should have been in 3NT. Should he have bid 2 clubs, (Stayman) on his first turn?

I would have denied with two diamonds, and he could then have bid two spades, showing five. I would have denied three with 2NT, and he could have bid 3NT.