I expect that the GOP will do everything in their power to claim that she did not legitimately become President. That's just what they do.
Sanders will be a footnote in history by then, whatever he does. He's badly squandered his goodwill right now and if he continues as he is now doing (I have every expectation that he will, but have a slight hope that he will somehow choose to act responsibly instead), he will have no credibility whatsoever left post-election among his colleagues. Even if he does formally "concede," any damage has already been done and that damage will NOT hurt Hillary. There are too many credible and well-respected surrogates (Prez O, VP Biden, various Senators including Elizabeth Warren, and other politicians, business leaders, union leaders, educators, etc., etc., etc.) who will be campaigning their hearts out for her.
Soon it will be too late for Bernie even to get on the Hillary train. I suspect that at least some are wary of what he might do if he did get on. He could continue to undermine her, as he has been doing with all the "she must do this and that and the other before I concede ....." He has lost a lot of trust.
But a concession might limit the damage to his own reputation among the people he still has to work with in Congress. He's not thinking too far in advance on that and karma can be a real bitch. He's not going anywhere except back to the Senate. There is no way whatsoever that any rational person would support having him head up a cabinet agency because those jobs are for people who have demonstrated that they can work with and support the 45th President, who will more likely than not be Hillary.