Not being able to capture kings is intended, you are meant to prevent Hectie from being able to make valid moves rather than make a direct capture. In most cases where Hectie is stunned and is threatened by one of your pieces, the game ends as a checkmate. The other issue that you're describing is probably an overlooked bug. My guess is that I forgot to make a "self-checkmate" check after a card is played, and you're in a position where Kate is threatened and can't make a valid move. I do hope to revisit this game at some point to make a small patch that adds a title screen, a few dialogue events, and fixes bugs, but I don't know when I'll have time, I've got some busy months ahead.
Charmers is mostly completed, but there is a chance of a small update coming in the future. It would probably just add text scenes for normal captures that don't have any, a proper title screen, and maybe some tutorials or quality of life features. In terms of animated events and other big things, the game is done, I don't think there's too much of a reason to wait for new additions.