I would invite him, assume he wasn't going to show and not bother making the changes to the dishes. I'd have something they could eat that I could pull out of the freezer, if they should happen to show.
This is what I would do.
Agreed, I would start with this. If you enjoy hosting otherwise, have one or two dishes on the side/in the freezer that could give him and his wife an adequate meal, then make the rest how you like. People with dietary restrictions (like me) who are reasonable, know not to expect a major spread of special food made by someone else who is also trying to accommodate many other people. At extended family Thanksgiving, for example, I usually just eat some turkey and a roll, and I bring my own food bars to eat later if I need to. I don't expect anyone to change or give up their casseroles, soups, butter-covered veggies, and desserts just for me.
An alternative is to suggest a potluck where everyone brings food. If Brother came, you would expect him to bring something he himself could eat. I don't know what his restrictions are, but perhaps you could also provide something which is a food everyone likes, but also happens to meet his restrictions. That way he'd have at least two things to eat, if he came, and you wouldn't have gone out of your way to accommodate him if he
doesn't show up.
So, that would take care of your irritation at making things special for him, only to have him cancel. But, it sounds like there is a second source of irritation, where you suspect he is lying about the reason for cancelling--sudden illness is a valid reason for cancelling, but if he knew in advance he just didn't want to attend, he could have said that and you wouldn't have gone to any trouble for him in the first place.
I don't understand why he would lie--is there some special pressure, perhaps from a parent, insisting everyone say they are getting together for Christmas? I think it's commonly understood that people have different obligations at the holidays and if he had just said, "Actually we're hosting a meal for some friends this year," I don't see why anyone would be upset about that or give him grief about it.
Also, to call out sick from your party, and then post photos freely on Facebook of another party he attended instead, is just about the height of dumbness. Is that how he normally is? Is it at all possible that the photos were taken earlier, and just posted on Christmas Day? Again, is there someone else--maybe an older parent who doesn't get on Facebook--that he's kind of hiding from? How does he act during the other get-togethers at the holidays?
If he says he will attend your party, then cancels that morning due to "illness" and posts evidence that he was actually fine and attending, even hosting, a party for others--yeah, that's extremely rude, childish, and hurtful in a way that's completely unnecessary. I'm just wondering if that's really what is happening, if that's consistent with the kind of person he is, or not. If he really
is acting that way, because that's just how he is, I'd probably stop inviting him to anything again.