Actually, the Invictus Games are a stand alone foundation with its own governance, partners and has pretty much run itself since its conception. Harry is the patron. It was nestled under The Royal Foundation of the Duke and Duchess of Cambridge and The Duke and Duchess of Sussex as an incentive of Harry's until that household split. I don't see why Harry would not continue as its patron even though he's no longer a working royal for the "Firm". Like Sentebale with Prince Seeiso, the Games are his incentive alone.
The Invictus Games are strong enough, IMO, to grow and prosper on its own and is pretty self independent.
https://invictusgamesfoundation.org/foundation/