The 4D version is not impossible in 4D space.
Since a Klein bottle is just a cylinder with its ends glued together a particular way, I think one way to sweep out a Klein bottle through time would be to have a rubber band split into two, then have each band travel along a trajectory that brings them back together with the correct orientation. The self intersection is easily avoided by having one of the bands travel faster than the other.
They'll still meet up again. It'll just take more revolutions to get there. That doesn't avoid it.