Cenell, if I'm reading you correctly it sounds like you're asking Google to provide a special type of Google account explicitly for automated testing your extension on Google sites in continuous integration environments. Is that correct?
If so, I'm afraid I can't be of much assistance. To my knowledge Google does have any kind of special account that fits this use case. I suspect that Google probably won't provide this kind of account because it would be heavily abused to bot Google services. The closest thing I can think of is OAuth 2 testing using a standard Google account and using the oauth-playground to get a long lived refresh token
as described here, but I don't think that's what you're after.
A potential workaround is to adjust your CI process. Rather than using a cloud CI system, you could have a single (or small number of) dedicated test devices. On these devices, you would use two separate Chrome profiles. One profile would be logged into a test Google account, the other would not be. You can use the Chrome CLI flag --user-data-dir=<path to directory> to specify which profile to load when launching Chrome.
Using dedicated test devices would allow you to avoid authentication/CAPTCHA issues at CI time as you would reuse existing sessions that have already successfully authenticated.
Simeon - @dotproto
Chrome Extensions DevRel