I actually think Charles will attend, at least some part of Invictus. However, I don't think he needs to, or should be. I just think Charles will recognise that not attending is the worse of two not great options. If he doesn't attend he knows that will be another nail in the coffin for his relationship with his younger son and his family. If he does attend, what is actually the worst that can happen? I suspect Charles would rather be at a very public event where their whole time together is videoed and public than spend half an hour with Harry in private. Charles - as most members of the RF - is well use to putting on a brave face, pretending to care (though I do think he actually will care) and being nice to people they might not want to.
The reality is, Invictus and the "will they, won't they" element of whether Charles will turn up will be on the front pages anyway. Charles IMO has not very much to loose from helicoptering in to Birmingham for a few hours and then leaving - perfect excuse -its Cam's 80th birthday which he must attend. That also gives Camilla a perfect out of attending.
I can see Anne (very sport and strong military links), Edward and Sophie (Ed has links to the paralympics) and even maybe the Gloucesters all attending based on their patronages - again for not very long. Plus Beatrice and Eugenie who may well be happy to attend for longer.
I actually have most doubt over Zara as she and Mike are very close to William. And of course, it goes without saying I don't expect the Wales to attend at all.
It also gives the Palace an element of control - if you want the King to attend there must be no leaks, no briefings to the papers etc. I don't know exactly how Invictus works but I get the impression a committee of sorts, somewhat separate from the actual charity that operates usually is in charge of organising the games in each place. Stuff that with enough military men and women who won't want to see Charles embarrassed and it might actually be possible to keep Harry somewhat in check and from doing anything embarrassing.
The bigger question to me is around what interactions, if any, Charles would have with Harry and family (if they attend) before and after the games.