Can I raise the floor of a culvert with concrete?
As I mentioned in a comment, a lowered pipe shouldn't cause a backup. The high point in the system is where the water level stabilizes.

That said, I see no reason why a concrete enhancement would cause problems on the intake side, assuming adequate flow volume remains. You wouldn't want to reduce volume on the outflow side as this could result in foreign object obstructions. 

Also, you'd use a standard concrete mix, and not an overwatered mix, which would be weaker and more prone to disintegration. It's not difficult to level concrete with a little agitation.