The short answer is I make updates work automatically by installing Chrome when I build my custom image (based on ublue/main and/or Bluefin). So it’s not an update in the usual sense. I just re-install it on a fresh container image instead of layering and having rpm-ostree handle the updates during rpm-ostree upgrade.
There seems to be an issue with the way Google signs its RPM packages that causes failures on updating. You can work around it by removing and re-importing the Google GPG keys. (I’m not sure why it works, but it does.) Unfortunately, you can’t do that on an atomic desktop with rpm-ostree, hence the need to build a custom image.
I have a script that can be called in a Containerfile or as part of a BlueBuild recipe that imports the key and enables the Google Chrome repo that already exists (!) in Silverblue. That allows me to include google-chrome-stable in the list of packages that I add to my image with rpm-ostree.