He wouldn't take the VP slot (for the same reasons as Hillary), whether he ran or not. It's a waste of talent, and seeing that he'd be the second youngest POTUS in US history (43 on Inauguration Day) if elected in '68, Bobby has in theory until 1980 to run. Keep in mind the man is very polarizing (think H. Clinton doubled), and a second Senate term would allow some bills of his own to pass, being among the 20 most junior Senators IOTL. LBJ said it himself IOTL: "I don't object to Bobby becoming President of this country, but not in 1968". Remember, the political aspect of their hatefest was not only about 'Nam, but about the cities, civil rights and various domestic issues as well.
Humphrey and he have a decent personal/political relationship, so it won't be blockaded like Lyndon did IOTL. But yes, he could run in 1968 or 1972 and win. One aspect that might be interesting would be an overhaul of the Indian Act- I think the picture with Pretty Boy in Nebraska tells that story.