Wouldn't a third party service repeatedly logging into my services from a LogDog IP raise some flag on it's own at the service's own intrusion detection?
I think that's a point you should be a bit more transparent about because I'm not sure there are a lot of people who want to risk their google account being locked because of your app. We are all aware of the customer support horror stories dealing with getting your accounts re-enabled after they've been flagged.