I think - and hope - that PW will nip it, pronto.
Whilst HAM reckon they are savvy, they are actually sadly lacking in that department. William and Catherine are in fact the savvy ones. They saw the American adventuress coming a mile off, that's why they were so concerned for Harry and tried to warn him, as documented by Harry himself. I suspect William won't have any of the nonsense that his father has effectively fallen for. There won't be any invitations to stay in Royal residences because William will be able to see that such approaches would be either manipulated by his brother or fall into complete farce. I imagine William sat with his team right now watching this melodrama play out saying "I said exactly this would happen."
I can see him being quite a radical king generally. He's very occasionally done pieces to camera, such as over the Panorama scandal - and I can see him doing that again, if he thought it was called for. He and Catherine very successfully tread the incredibly fine line between regality and relatability. I think they'll continue to nail it, and leave HAM looking even more like ridiculous relics from a past era.