What is your neighbor asking for? The removal of the lift? What are they alleging you did illegally? I don't really see a case as you obtained all the required permits and jumped through all the hoops. Usually they will drop the case through mediation or with some cash "compensation".
Try contacting one of the MAJOR dock and/or lift manufacturers. Especially the one that you used. They may be able to provide expert testimony. Once again, there will be some "compensation".
Instead of trying to argue why you need the lift, why not PROVE that you obtained all the necessary permits and that it's legal. If the neighbor has a problem with the appearance they should be filing a complaint with the local building inspector. My guess is that they did and hit a dead end. Something to look into as it would be a public record.