Rule of thumb for litter boxes and cats is "one more box than cats". With three cats, four litter boxes is the norm.
Adding two more boxes may help the cats co exist better. The boxes will be cleaner, for one thing, unless you are scooping every time they are used, which would
be a chore, LOL
I actually do scoop as soon as a box is used if I am home to do it. However if you are like most people you scoop only once a day, and having to pee or poop over another cat's pee or poop can really rub a cat the wrong way. They may be happier in general with more choice of bathroom facilities.
Adding more elevated places for the cats, since they fight over the existing one, may help. Cats like to be up high, instinctively.
Feliway can be gotten relatively inexpensively through amazon.com or other places with discounts and coupons (I have a friend who just purchased several refills at $7 each).
Of course for it to be effective you have to have enough going, depending on the size of your home. So yes, it can get costly.
And granted, it won't help in every situation. I love it for stressful situations, and I know others who have had good results with it, but I have only a very small apartment so only need one. And I don't keep it going all the time, just when needed.
You might give the rescue remedy a try.
Otherwise, it sounds like Red isn't suffering much stress from his dislike of the other household pets, and the other cats have each other to bond with right? And the dog of course has his walks to keep him happy. Hopefully you are training the puppy to leave the cats alone.
Cats sometimes become more mellow with age, so you never know, someday.....