Hello and welcome. My name is ***** ***** I have been in the dog field for 25 years. I am a Therapy Dog & AKC Evaluator/trainer/ as well as an assistance/service dog trainer, specializing in behavior. It will be my pleasure to help you today.
It would be a good idea to have your Vet do an examination of the throat to make sure that all is okay and have them prescribe medication if needed. Only a hands on exam can determine if there is cause enough to put the dog on any meds. Dogs bark at other dogs all day long and do not develop laryngitis so you want to be sure nothing else is going on which only a hands on exam can give you.
As for the barking at the fence. You can block the stimuli by covering part of the fencing where they connect or see each other, you can try a barker breaker in the yard somewhere which will go off with a loud sound that is irritating to some dogs and so they may stop barking, or you can build a dog run pen for exercise away from where the other dog lives and block the stimuli on that fencing. They do make shock collars but the also make collars that spray water or citronella which can have a dog stop barking.
You are seeing fence aggression which it very normal in dogs. However how it is effecting your dog's health is something you want to take care of since there are vocal changes.
IF the dog knows the command leave it then I would use this each time he goes to the fence to bark. That is a simple solution but you have to be home in order to do that. Knowing that there is a problem though, it would be best to accompany your dog on outings when the other dog is out.
A barker breaker can be purchased on amazon, and I would get one that can be left outside for when you get busy. This may also stop the neighbor dog from acting out as well. it is like a self corrector when the owner cannot be there.