Is it $NP$-complete to decompose bridgeless cubic bipartite graph into edge-disjoint paths of length 3?
As pointed out by David Eppstein when I asked a related question a while ago: you can do that in polynomial time. 

The idea is to find a perfect matching $M$ (which exists and can be found in polynomial time since your graph is cubic and bridgeless), and then to use the edges of $M$ as middle edges of the $P_4$'s.

To complete the construction, remove the edges of $M$ and orient the remaining cycles arbitrarily. Then attach to each edge $\{u, v\}$ of $M$ the arcs going out from $u$ and $v$, and you are done.