You could try contacting Itch and see if there's something you can do or some setting they can change. I don't really know if something can be done just by changing some setting or not.
On the other hand, I would think that removing the demo from the page and moving it to it's own separate demo-only page might work.
I'll be honest and say I don't know for sure because I haven't seen a dev try it. And it is kind of dumb to have to put a demo on a separate page and link to it from the full game page. But I would be curious to know if that makes the full game permanently claimable or not.
I think it would work however, and I'll explain why in two ways. Basically, it appears that what specifically blocks something being permanently claimable is having no minimum price for downloading one or more files. If there are no free files on the page, you can claim the full game permanently during a free sale, in every case I can recall.
The more complicated way of explaining is that "claiming" something on Itch seems to work in tiers. You only get permanent access to the lowest price tier on the game page, when claiming during a free sale.
For example, if there's only one price tier (no free files) claiming gives the full game permanently.
If there's two price tiers (tier 1: free demo or other free files, tier 2: paid game). Then claiming gives temporary download access to tier 2, which only lasts as long as the sale.
If there's two paid tiers (tier 1: full/normal game, tier 2: higher price point to pay for extras or dlc, etc). Then claiming should give permanent ability to download tier 1. Claiming would probably also give temporary access to tier 2 for the duration of the sale too, but that's much more rare so I don't know for sure. I strongly suspect that's how it would work though.
So in short, it's the lowest price tier, and only the lowest price tier which becomes permanently downloadable when claimed. if the lowest tier on the game page is free, there's no permanent access to anything paid because that's a separate tier, which is above the lowest tier.
I hope that made things more clear.
Itch should REALLY have this in their documentation somewhere easy for devs to find. It's unintuitive because it's not like other stores, and most devs and most itch users have no idea that it works like that.