I am mean-spirited enough to believe bad things about just about anyone but this doesn't really add up.
There are plenty of eye-witnesses to King Charles being absolutely devastated when he went to bring Diana's body back from Paris. And he was not popular at the time and people did want to blame him, so if there was any sense that he was anything other than devastated and grieving, people would definitely have picked up on it.
Charles Spencer is adamant that King Charles did say what he alleges.
The only logical assumption I think is that King Charles did say this, or something very like it, but did not mean it in the way Charles Spencer took it.
My theory is that Charles was thinking about his Uncle Louis Mountbatten: killed in a shocking way, a massive news story, huge funeral, but largely forgotten by the public by 1997. I think Charles was thinking that Diana would go the same way and be a footnote to history soon enough so they might as well have a big dramatic send off.
If King Charles really did say "we will forget her soon enough" that does sound callous though. I suppose he could have meant "we" as in the nation, but I suspect he might have said "she'll be forgotten soon enough" and Charles Spencer has remembered it differently.