I don't necessarily mean this very second. Perhaps in a few years because he's not in good health or too old - I don't for a second think he would abdicate because of Charles Spencer's nonsense. I don't think he will abdicate anyway, but this is just hypothetically speaking
The King Charles haters seem to hate the whole BRF and have it down as a mafia organisation that pretty much killed Diana. Charles Spencer is I suspect very much a believer in royalty, but his (retrospective) bitterness seems to start with the future Queen Elizabeth marrying Prince Philip. He even writes in his book that there was a romantic connection between his father and a (preteen?) Elizabeth but Louis Mountbatten put a stop to put - the implication being that John Spencer and Elizabeth would have married but Louis Mountbatten did all he could to end their romance so that his nephew, Philip, would marry Elizabeth. Otherwise, there would have been a Windsor-Spencer dynasty and the son of John Spencer would be king today!
So, I guess on a superficial level, if King Charles abdicated Charles Spencer might think it's a win for him, but with William as King he would quickly have to reorient his attitude to the royal family and be nice about it (he made a bullshit post on Instagram a couple of years ago about how wonderful and graceful Catherine is for being brave in her fight against cancer). Even if he did that, my suspicion is that William's feelings towards Spencer are not at all warm, especially now Spencer has put out this book exploiting his mother and attacking his father. And if William hates anything it's these salacious media scandals.
A King William may be superficially polite to Charles Spencer but he would keep him very much at arm's length, especially if old King Charles were still alive. Even when old King Charles dies, Charles Spencer will be pretty much frozen out from anything to do with the royal family, even if his nephew is head of it.
As for the King Charles haters, they will be confused. They will probably say of King William when does something good that "he's just like his mother". They will switch to loving Catherine (right now, many don't) and will go on about how much George looks like a Spencer. Many of them aren't even British and are just leftover Diana fan girls from the 90s who don't really understand what the royal family is.
On the whole, however, if they want to support Diana's legacy, then they will have to support the British royal family under King William.