I have started looking into this and have some thoughts which I'll share here. You don't need to take these into account, but always positive if you can comment - constructive criticism is always welcome
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1NT - (3♣): same system as over our strong 2NT, where X = Puppet.
My consideration here is that, since the 2-level contracts are no more available, hands within 8-ish HP (not AA or AK) would pass. However, I am still thinking about when responder sits with ♠ values and would like to X for penalties.
1NT - (3♠): this is a hell of a problem to fit both 4/5 ♥, asking for a ♠ stopper and minor(s).
What about having a forcing pass here, where a weak responder would take out in his 2nd round?
1NT - (p) - 2♣/2♦ - (3♠): this will be easier since responder has already shown his length in ♥.
But what about when you sit with 6♠4♥ and invitational values and it goes:
(1♦) - 1NT - (p) - 2♥
(3♦) - p (denying 3 ♠) - (p): what is your 3♥ now, natural 5-4? X = re-transfer or penalties? Or maybe pass by opener should now be something else than denying support in the shown major?