(If the Mayor is a 'friend' of William, and doesn't want to be drawn into the 'civil war', why would he invite Harry and wife to be guests of honor at his party?)

I seriously doubt they would have been the 'star' guests at the party. That's the Daily Fail embellishing things a bit. As to why they were invited, they were invited because it would have been the height of rudness not to invite them. It was not a private party, but a Networking party. Sadiq Khan was there to promote and network for London. It was a very public party. The invite from the Harkles was for a private meeting, something to be avoided at all costs by a public figure like Sadiq Khan. Another bit of embellishing by the Daily Fail is the friendship thing with Prince William, more of an acquaintance than a friend. Sadiq Khan is a politician. Royal are not friends with politicians. Friendly with, yes. Friends, no.