let me get this straight. he lets the dishes pile up goes home then comes back in the morning and squeakes they havent been done and hes the dishwasher?
ignore him and let him make an official complaint to the management.
If they then tell you its your job to make the dishwashers job easier by DOING HIS JOB for him ask for a damn pay rise.
Im sorry but for me personally I wouldnt put up with it unless this guy had pulled MY bellybutton out of a hot LZ in 68

if i was you id merely say to the management that you are a hard worker and you do your job as asked, but to be moaned and squeaked at by another co worker for not doing the dishes that was actually THEIR job is unacceptable.Tell them you'll help out as best you can at ALL times but you wont stand for this guy moaning and squeaking at you.
You have rights as a worker.If it looks like this stance might lose you your job and you cant afford to then button your lip and just get on with it until you can afford to leave but make sure they know you wont accept unfair dismissal.
in the mean time record some sounds from appocolypse now onto a hand held tape player and when hes alone doing the dishes play it !!

as soon as he looks for where the sound is coming from stop it and act like you heard nothing lol. Hell at least you'll have fun doing it

mind you in all seriousness if this guy is really a war hero and not just some army clerk who served out there maybe you should allow him his few perks of the job? after all its really thanks to those that give up their youth to fight for what we believe in that we all have a job to go to.We do 'owe' them.