Hopefully my response will clear up a few misconceptions that seem to be around:
1. If you play "standard" NMF, then 2S does NOT establish a game force. It shows 12-13 and 3 spades. With 14 and 3 spades, partner skips to 3S (forcing).
2. That means that unless you have agreed otherwise, 3s is NOT FORCING. It is an invite. I don't think that's what you want to do with this hand. If you don't believe me, perhaps you will believe Larry Cohen:
https://www.larryco....nter/detail/108
3. 3C isn't typically played as forcing, either. The meaning of 3C depends on how you play 3C directly over 1NT. If you play that as a weak 46 (4 spades 6 clubs), then 3C now is a 45 or 46 invitational hand. If you play 3C over 1NT as invitational, then 3C now is a weak 46 and is to play.
Otherwise, you have no way to show the weak 46 hand and the invitational 45 and 46 hands.
4. I probably would bid 6S at this point, especially if this is MPs. Partner didn't bid 3S, so you are highly unlikely to have 7S. But you have a 5-loser hand with a 9-fit opposite an opening bid; that ought to make slam. The advantage of just "blasting" is that the opponents have less information regarding what to lead. They might have the AKh between them and fail to lead H.
In addition, if you play standard expert cue-bidding, it's hard to find a control in the suit below the trump suit, as in a cue-bidding sequence, a 4H bid is not a H control, but instead is "Last Train."
Finally, the reason why I say "especially in MPs," is that in MPs, there is an additional reason to blast. If you cue-bid or use key-card and end up in 6S with partner having the Kh, I guarantee you that LHO will lead H even if he doesn't have the Ace. This could cost you a valuable overtrick.
5. This sort of hand is the reason why NMF is not a great treatment. Two-way NMF or xyz is a lot better. With two-way, you would bid 2D, not 2C, to show a game forcing hand (2C would show game invitational strength). Partner would bid 2S with a minimum and 3 spades, and now that you are in a GF auction, 3C would indeed start a cue-bidding sequence. Much easier.
Cheers,
Mike