Hi,
Thanks for taking in two rescue dogs! As a long time animal rescuer and foster parent for numerous animal organizations I've had experience with diffusing animal aggression. I've only taken in large breed dogs and the Humane Society often calls on me to foster animals who have suffered from abuse and cruelty seizures. Sometimes I've had as many as four male dogs living in my home. Another thing I did for many years was educate the public on dog behaviour. Here's some things you may want to possibly try. These are only my opinions and experience so I'm not going to be insulted if anyone doesn't agree with some things I'll suggest.
First, I'd get the one checked out by a vet if you haven't done so already to verify that yes, he is indeed neutered in the true sense as his hormones can still be racing throughout his body. At dog parks in our area intact males are not allowed in as other dogs view them with jealousy and they themselves have more pent up aggression than a neutered dog. Once neutered ,it takes a few wks for the hormones to dissipate within their body.
Second, I agree that food, toys and attention can all be triggers and should be avoided. With me, I never let my guard down and always watched for signs of stress/anxiety and diffused the situation. Agreed it's a little tougher when they are outside and you can't always view their body language. A tell tale sign is their tails...if a tail is being held straight up the dog is saying, "I'm the boss" if the other dog also has he tail up he too is saying , "Nope I'm the boss" and a fight will probably start shortly. If one lowers his tail then that one is backing down. Ears are also a good indication of a dog's mindset and there are probably some good sites on the web that can teach you dog body language and I feel that will help you anticipate their feelings towards one another at a particular moment and diffuse the situation. For me in a house that meant distracting by various means but could be something simple as calling the name of the one who is in an aggressive stance and allowing the other to walk away and not maintain eye contact with the aggressor as that is another tell tale sign.
Thirdly, Very often it's not recommended to have two dogs from the same litter grow up as adults together, nor two males and here's the reason why. They become so focused on each other that they grow up being more dog focused than people focused if that makes sense to you. In other words, you should be the alpha over them. I've broken up fights not by getting in between but by saying Leave It! and if they view me as alpha they have stopped. Being alpha to your dogs is not about force but by your position. A hose by your side is also a way of separating two fighting dogs but never pull apart or get in between as your safety is of concern. Even in world of wolves they will oust a leader that is too forceful but having a strong leader makes them feel more secure. Saying that I only am mentioning it for others to possibly avoid doing(getting two puppies from the same litter at once) and hopefully not insulting you by doing so.
Most large breed dogs reach maturity at 18 months (smaller dogs reach it sooner). When dogs, rabbits and many animals reach puberty they will fight their litter mates when previously they got along just fine with them as youngsters. Dogs that are going to be used as aides only start their serious training after the age of 18 months as previously they still display puppy like behaviours and mentality. Starting at 2 , large breed dogs will stop growing but grow larger in their chest area..they buff out.
Dogs live by a code of law believe it or not and these are some of the things they live by. Please note - just like people some dogs will break this code.
Most adult dogs will never attack a puppy as that goes against their code just like in our people society adults have the code not to harm babies and children. Adult dogs do however discipline puppies and that is not to be confused with aggression against them as they are teaching them good canine behaviour and should be allowed. If prevented you will have a pup that thinks it's okay to go around and pull on another dogs ears and doesn't learn bite inhibition. However, once they hit maturity one dog will usually try to place another dog or others beneath them in their hierarchy. Some don't fight but do so with body posturing - hense watching the tails, ears and body language you can tell which is the alpha in the household. Some dogs have no interest in fighting and will take to the leadership of another willingly by displaying the lowering or wagging of their tales.
Males generally as a rule don't attack females as that too is against their code. However, if they do it's usually when one or the other has had enough and will put the other in it's place. It's why most animal societies will recommend you should adopt the opposite sex dog should you have one already in your home.
A truly aggressive dog will be just like a person - "kick a man when he's down kinda thing." An aggressive dog will keep attacking a more submissive one or one that has shown his belly to show in the dog kingdom he acknowledges the other one is alpha. Alpha dogs unlike aggressive ones will allow the submissive one to get up when he shows his belly. They will also not draw blood (this is a good sign with yours as they both appear to be fighting over their status but are avoiding all out aggression).
A common saying amongst those that work with dogs is a good dog is a tired dog. Good advice from some who mentioned that taking them for long walks or allowing them to have fun is a good way of diffusing any pent up frustrations. Do you have someone that can help you with this? Very often when I introduce a new dog in my household I never bring them home immediately as dogs are territorial by nature and always do intros on neutral territory first. This isn't necessary in your case as they currently both live with you but mentioned it for those wanting to bring home another dog into their household.
I would have someone help you and walk the dogs on leashes opposite one another on a sidewalk or a short distance away from you in a park perhaps...then slowly you walk closer to one another. This doesn't happen immediately and again watch for signs of frustration but eventually overtime you can walk side by side with the dogs. It may not happen but it's worth a try and important to remember that for every step forward ..sometimes it's a step backward but if you're willing to put in the time you can change behaviours.
Dogs that learn to enjoy themselves and have fun away from what they view as their territory can then be brought back to their environments can develop a friendship and mutual like for one another away from the stress of feeling they had to guard their territory. I use to say, Imagine if your hubby, parent , wife said to you , " I liked you so much I had to get another" you wouldn't be thrilled to find what you viewed as your replacement in your home and neither is your dog if the introduction is not made properly. (That part doesn't apply in your case but for those that seek to get another dog to be a pal to their current one). It's why Humane Societies always insist your bring your dog with you with seeking out a pal for your current pet- they should choose their new friend and they meet on neutral territory to see if they get along.
Lastly, you might consider calling in a dog behaviourist to give their first hand opinion as to what they view is happening with your dogs. Granted it will cost you but I always use to say whatever you put in remember that you will be dealing with this behaviour (whatever it may be) for the next 10-15 years and initially the work or cost in the beginning will be well worth it as an end result. I wish you all the best and feel free to ask any questions you may have. Again, this is only my personal experiences and opinions based on the dogs and other animals I have living in my home.