There was enormous momentum for the CRA in both houses of Congress and in particular in the Senate even before JFK's death. There was also a huge public upsurge of interest and support, especially in the aftermath of high-profile incidents of disobedience and violence in the south and the March on Washington.
LBJ's role in the CRA is overstated. Mike Mansfield, Everett Dirksen, and figures like Hubert Humphrey all played an enormous role. Even if you posit that it wouldn't have passed till after 1964 without JFK's death, remember that the election results would still have returned an even larger Democratic majority (even if it was short of OTL's post-1964 numbers).
What is possible is that other post-CRA bills like the Voting Rights Act, Fair Housing Act etc don't pass or are watered down. Though as David T points out, the Supreme Court will likely push some of these issues itself, and Kennedy's DOJ might itself aggressively push action through executive and administrative moves.