Let's see if I can guess what you've thought of...
-Fixed point rotation
-"On rails" rotation
-Camera "switching" depending on gravity (like, a fixed camera at each spot needed--I feel like that's one of the first things you tried, though)
-Curved/square path camera (Also, I feel like that was an early thought)
-Series of cameras that "jump" to each other as the path progresses
-Totally stationary camera
-Start and end cameras
-"Follow me"-style cameras (I think the official term is "over the shoulder")
-Isometric angle camera (only one that looks down on the whole path at an angle)
-Whatever it is that Google Sketchup uses for its orbit tool (or 3DS Max, I guess; they might be the same)
-Sonic Adventure's idea of a "free camera" (minus the infamous issues with it)
-Camera "collision," where having a set of invisible walls bump it into the right spot when it hits them
-Some kind of mirror effect until the path ends
-The style that Super Monkey Ball uses for its replays (where you can change the perspective, except in this case the switching is done with trigger points instead of a button--but, again, that's probably another really obvious one)
-Inserting a straight up cut-scene that looks identical to the gameplay (based on the above response, I'd assume this failed, too)
-Stringing together hundreds of images (one per frame) of each position on the path @ 60fps (This might very well just be a bad idea, though)
-A purposeful rendition of Sonic 2's "The camera's too slow to keep up!" in Chemical Plant Zone
-Some kind of motion blur effect to hide the camera's screwiness
Bearing in mind that I really have no clue about how game engines implement cameras, nor do some of those make any particular literal sense, I'm probably the last one to be trying to figure anything out. I'm just hoping that maybe on an off chance something can trigger an idea that'll be feasible.
That being said, sometimes just a break can do some good. Never hurts to look at something with a fresh/revitalized set of eyes!
Edit: By the way, those renders for Sonic-3KD look stellar!