Interesting... though I'm not sure I have that level of trust... have you considered trying to integrate with a secondary browser surface, like the chromium browser that playwrite/puppeteer load?
You can have the app load in the integrated browser, and you will already have dev channel access, just need to side-load your extension into the "temporary" browser. This way users don't have to trust AI extensions "inside" their local browser. Just a thought on this.