What is wrong with this reasoning that finding the genus of a degree 3 bipartite graph is NP-complete?
You cannot reason in this way. Try proving it formally to see what goes wrong.

Also, consider the following example (which is admittedly slightly different than yours). Chromatic number is NP-complete, even for planar graphs. The 4-coloring problem (i.e., the chromatic number problem with the number of colors fixed at 4) is NP-complete. However, the 4-coloring problem for planar graphs is in P.