Remote Chrome WebDriver on gcp

I run remote chrome webdriver in GCP Cloud Run. It works well if I use unathenicated service call and i can connect from my java code to this Chrome Webdriver. However, once i enable gcp authentication service-to-service, it stops working. I wasn't able to find a way to provide google token to Selenium Remote Webdriver. I understand that I need to extend one of existing Selenium java classes, like RemoteWebDriver or HttpCommandExecutor, but it's not clear which class exactly. Please advice.