From what I remember, I don't actually think the King is in the books. I know Charlotte is 100% not in the books so I'm assuming the King isn't either.

Also, Bridgerton does take place during the Regency Era in which the King's eldest son Prince George was Regent when the King was suspended of all his royal duties so maybe no one really considered him as their King anymore especially seeing that he quite literally went insane following the death of his youngest child. But again Bridgerton isn't really historically accurate and is completely fictionalized based on facts so who knows. They might bring it up in season 4.

I think that it's interesting that the King's actual reason for being mentally unsound isn't really mentioned further, other than Charlotte saying that one child died years before, but not really saying more. If you watch "Outlander", as I do, then it's clear that both shows are in the same time frame, but it's shown very differently in "Outlander". I think that maybe this series isn't as focused on the history aspect of the time period as much as it is on the different characters and plots that are developed for each character. I think that "Outlander", aside from the time travel aspect, is really more focused on the history as well as the characters. I say this because season two and season three, are focused on the Battle of Culloden and the aftermath of what happened during that battle.