The thing is, one of his strongest messages has been about bringing people together and healing division and reaching out to opponents and so forth, and inviting Clinton on board would be seen as a way of doing that in the context of the Democratic Party.

There'd be counter arguments around the claim of Clinton being divisive in herself in relation to the wider electorate, but binding in the Clinton wing of the party would make a lot of sense.

It'd make less sense for her to accept it, especially given her age. (Though even after eight years she'd be younger than McCain is now...) My bet is if he gets to be candidate he'll offer her the job, and she'll decline it.
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If Clinton does get to be the Democratic candidate for president, one theoretical possibility would be that she could even pick Bill as her running mate. No rule against a former President being Vice President that I can see. And nothing to prevent him from taking over as President if she got elected and couldn't carry on, or whatever.

There's no bar on being President a third time, just on being elected to the office:

Twenty-second Amendment to the United States Constitution Section 1. No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once...)