I am also Beeminding a job search, though I am using @adrienle’s Beeminder-Toggl integration to keep track of time.
I sort of assumed that after accepting an offer I’d just count time at work towards the existing job search goal. Also, I’ve heard elsewhere that it’s important to quietly keep looking after you accept an offer, because there’s still a chance of things falling through until you really start.
But I think emailing support for all these sorts of exceptions could work even better. Support is the best!
I just like to spare them unnecessary communication because of how or how often I mess up data points or forget to enter data because I got distracted by a conversation about Mars rovers or blue whales around deadline time. Yeah, sometimes it’s been like 20 times a single week for stuff like that. 
But they are always very chill and kind about it. 
You should definitely not hesitate to contact them when your amazing work has culminated in a new job!
Very best with your job hunt, fellow job hunter!