Unfortunately browsers need at least a monthly security update. Bromite doesn't have a commit since January, so it's dead. And the last release is from December. Even if for some reason it didn't have that specific webp vulnerability, it has 11 months of other security issues. You must stop using it immediately right now
For a browser there's no data to transfer except the few bookmarks and the opened tabs (you aren't using a dead browser with no sync as your only password manager, right?). Install a new browser, then share the bookmarks and tabs one by one.
I don't understand how root can break your Pixel (it doesn't void warranty) but anyway unlocking the bootloader for rooting requires a full wipe and all you can get is a config database that could only be used with bromite (or forks, if they didn't change too much)
For your next browser choose one that has hundreds of devs as staying behind updates is a massive task and a small team (or a single person) like the one behind bromite can easily burnout and disappear overnight.