Flashback was, no IS such a great game. I just started playing it on my iPhone since they allowed emulators.
I can't answer your question but I suspect you're correct that the two behaviors don't play well together.
I've also thought about trying to replicate the same movement and think I'd have to build it from scratch rather than using any existing movement behaviors.
I don't think platform behavior would work because in Flashback, movement is quantized whereas platform behavior is continuous.
And tile movement doesn't deal with jumping/falling. Maybe you could disable tile movement during jumping/falling and move the sprite 'manually'?
EDIT - just found this:
spriters-resource.com/fullview/54385
Also not sure custom movement behavior would help as it also look continuous.
But I'm just rambling so not sure - I think I'm going to try this myself and would love to see how you get on.