Seems unlikely.
For starters, while the Chiefs won't be in a great spot for the cap in 2022, I don't think they'll be in a terrible one either, and they'll easily be able to find a way to keep him. The franchise tag will certainly be an option, and while it's going to jump up considerably from 2021, it'll like be in the $16-17M range for 2022, which is certainly a fair price for Brown. So even if there's questions about his ability to play the position, they can basically keep him for 2021-2022 at a combined like $20M, which is a bargain. They'll probably lose a guy like Mathieu next year if they keep Brown, but I think its pretty clear from how they're structured that they're willing to invest pretty heavily on offense, maybe pay like 1-2 guys on defense, and then go draft heavy after that, which is what I expect them to do in 2022 and going forward.
That being said... if they did opt to not keep him, it really wouldn't be the end of the world. They didn't give up a ton to acquire him in the first place, and if that's a one year rental, so be it. They're probably only surpassed by the Bucs as teams who are in as big of a "win now" mode as it comes, so it's not surprising to see them throw some draft picks at a guy who can protect the future HOFers blind side, if only for a year.
I don't hate the trade for either team. I mostly hated it because we traded him to the Chiefs, which is almost certainly a team we'll need to beat in order to get to the SB. If we had traded him to even somebody like TB or really any NFC team, I'd have no issue with it at all.