I did need to add that detail.
@Ovinomancer suggested that a successful roll would be retroactively justified, so without that detail, a success to try and get him to burn down the orphanage would result in the Captain wanting the kids to die for some reason which he would come up with. With it, the "success" won't be successful.
Under this new scenario, if the PCs have evidence, I would give them a roll to convince the Captain, or if the evidence is really strong, not even require a roll. Altering the scenario alters the point that I was trying to make.