They are in an odd spot of having extreme de jure power (Charles could disband parliament tomorrow) while having a completely unknown de facto power (would they listen? It didn't work out too well for another Charles).
King Baudouin of Belgium tried to exercise his power in the 90s by refusing to sign the abortion law, citing his catholic faith.
The law got signed in to effect anyway. The "trick" was that parliament can sign laws in to effect if the king is incapacitated. They declared him incapacitated, signed the law, and declared him "capable" again the next day. Whether that was truly legal and in accordance to the letter of the law was a matter of some debate (in my reading of the text, it's not), but I expect things will go in a similar fashion for the British monarchy if Charles really tries to use his "hard power" (rather than "soft power"/influence) in any way.
Keep in mind that king Baudouin requested himself for a solution where he would not have to sign the law without I obstructing the democratic process. So it is not fully true that they went against the will of the king.
It seems as if in Elizabeth's life (if "The Crown" has any bearing on real life events) she was able to wield some amount of de facto power either due to her reputation/prestige personally or from her role as monarch.
> Charles could disband parliament tomorrow
Elizabeth had some amount of real world power perhaps via the potential threat of this.
