RFK is also up for reelection in '76, but with his longtime ally Hugh Carey in the Governor's Mansion the Legislature probably rams through an LBJ law. He would probably ask Carey to appoint Moynihan as his replacement.
In the Senate, Mansfield is stil Majority Leader unless he retires earlier. Either Ted or Robert Byrd become Majority Leader in that case.
He'd have the opposite relationship that Grover Cleveland had, then. Good relationship with the Senate, bad relationship with the House.
The Speaker would have been Tip O'Neill, and while I'm sure O'Neill would have tried his best to accommodate the President's policies as a party man, he would have held a grudge against RFK's staff and RFK himself for essentially being the "Irish Mafia" that kept Congressional leaders out of the loop during the JFK Administration.
Off topic, but there was this famous joke that when a thief came into the White House, Frances Cleveland told her husband "There's a thief in the house!" to which Grover sleepily responded, "Maybe in the Senate, honey, but not in the House!".
Just felt I had to share that

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Note: The Irish Mafia was made up of Kenny O'Donnell, Dick Donahue, and Larry O'Brien, and the like. As you probably know, they were Kennedy's campaign advisers and officials in his administration.