How do you think Vietnam would go under either a Humphrey or Kennedy presidency in 1968? Yes, for an upcoming project. One rule: NO "Kennedy unilaterally withdraws a la McGovern '72" on Jan. 21, 1969. I'll chime in later. 
Agreed. I think Cambodia might be left to ARVN/VNAF. RFK will be very tough on Thieu: IOTL Thieu hated Kennedy's guts and allowed the Saigonese press to depict him in a NVA uniform. Rest assured that would not be forgotten.Kennedy will likely try to persuade Thieu's Asian allies: Lee, the Tunku, McEwen, Holyoake and Marcos, to pressure him as well. If Kennedy pursues a tougher line, he'll have to keep his Vietnam policies afloat with Republican votes.
Humphrey: pretty much the same, but willing to authorize a Cambodian incursion by US forces.
I think that all three would pursue essentially the same goal. Kennedy and Humphrey often spoke of "peace with honor". Like Obama on Iraq, their remarks were (deliberately) sufficiently vague to enable a wide variety of interpretations, depending on the audience. Or as Kennedy put it when asked of his own position: "whether you're a hawk, a dove, or a dawk or a hove..." But yes, if nothing else, neither Humphrey or Kennedy want their positions undermined or ego bruised by a "fifth-rate power" as LBJ called DRV in 1965.
If Kennedy wins, then he'll right (pun intended) the economy in time for a repeat of 1964 in 1972. So neither really need the antiwar crowd in the long run.
How do you see Bobby "rightying" the economy?
Also, how do you have Bobby winning the 1968 election?
Agreed on Humphrey and Kennedy. If you asked Humphrey what the policy differences were between LBJ and himself, he'd say "umm... there aren't any".
Getting back to your question about Church-Case: HHH wouldn't be in office after 1972, it would be Reagan, which is a different story. SoDems/GOP would kill it in an instant.
Kennedy will have to get near-unanimous Republican support in the House, which he'll get. But in the Senate, he'd have to veto it and the Republicans would prevent a veto override. If Kennedy twists enough arms, he might get the SoDems to prevent initial passage. The best thing that could happen for Kennedy in 1970 and 1974, both on domestic and foreign affairs, is a SoDem-GOP majority. Like Clinton, he'll find it much easier to work with them than the liberal Democrats, who are stuck with him anyways.