The Overfull conjecture in graph theory and $coNP$
For the possible reduction you are mentioning, if you find such a reduction from SAT to this particular edge-coloring problem, and additionnally you assume OC, then it would indeed mean  NP $=$ co-NP. However, it would be very surprising:   it would mean that basically, up to encoding, you found a way to always provide a short certificate that a formula is not satisfiable.
This seems quite unlikely to reach with a simple construction.