There exists a somewhat analogous situation to this in lizards, with 3 males that have different appearance, though they are not so drastically divergent as to be unidentifiable.
In this case they system works, and stays stable, because they are playing the evolutionary equivalent of rock, paper, scissors. Each male 'beats' one other males morphology when it comes to competing for mates, but they are equally beaten by the third male. Every generation one of the male morphology will have had more success then the others in reproducing, lets say your α-male, leading to many of α-male existing in the current generation. However, since another type of male, lets say β-male, is better at securing mates when competing with the far more common α-male these β-male will secure more mates in the current generation, leading to more β-male in the subsequent generation. That generation, now full of β-males, will in turn see the γ-male successfully out compete for mates with the common β-males. This finally sets the α-male up to make comeback in the subsequent generation since they can easily out compete with γ-male.
In lizards these males could be looked at as standard, harem, and sneaky males, based off of their mating strategies. The 'standard' males are exactly that, normal sized males that will compete for rights to mate with each female as it finds as most lizards do. Harem males are larger then standard males, and can easily best standard males in a fight. These males collect a large number of females together into a harem and guards them. When standard males are common the Harem males can easily beat the weaker harem males in fights, allowing them to secure large harems with minimal competition and thus secure numerous matings, effectively out competing Standard males.
Sneaky males are in turn the kryptonite factor of Harem males. These sneaky males are weaker then both other types of males, but they are fast and have a different mating strategy. Rather then fighting the much stronger Harem males they allow the Harem males to collect females into one place and then sneak into the harem to mate with the females while the Harem males are distracted defending their harem. When Harem males are common these males spend most of their time competing with each other to try to secure large harems, leading to increased injury and death amongst harem males and lots of females in one place for the sneaky males to mate with during the distraction. This results in the sneaky males producing the most young in the subsequent generation.
Unfortunately for sneaky males they can not defeat the other two males in a competition. Once sneaky males dominate they will not be able to find harems to sneak into, and instead will face standard males who can defeat and drive away sneaky males to secure mates, but never collect a large enough harem to allow a sneaky male to secure a mate while they are distracted.
A few different species of lizards demonstrate this pattern, and it has proven evolutionary stable. Males of every morphology secure mates in every generation, but the fact that each male 'beats' another male in competition prevents any one type of male from winning the evolutionary arms race, the more common any one gets the better off their counter male morphology becomes.
You could do something similar with your males. The only catch is that you want your males to be significantly different, while the above males have a less drastic morphological difference. My suggestion would be to make the males niche be more then just mating strategy. Each male is evolved for a specific niche of habitat or prey as well, developing more drastic changes in morphology to fit that niche; but still keeping the concept that one one morphology grows more dominant another will benefit from it.
One example I can come up with is that they prey on different levels of the food chain. α-male are larger, and eat a larger carnivorous species. β-male are smaller and eat a species that is preyed upon by whatever α-males eat. Thus when α-male are common there is more food available for β-male's because the β-male don't have as much competition for resources now that α-males are eating the other major predator of β-male preferred prey. In turn perhaps γ-male evolutionary niche depends on the prey species of β-male being less common. Maybe γ-males prefer some resource or living condition that the prey of β-male also prefer so that they can flourish once β-male remove their competition.
perhaps γ-males in turn build damns that change the nature of river above and below the river, creating habitat changes that make the species α-male prey on more common to complete the rock-paper-scissor dynamic.
If you make it so that these males have evolved to benefit from another type of male being common in multiple ways, benefiting in mating strategy as well as niche or presence of their prey species, you may be able to create a niche that is unique enough for each male to justify radically different morphological, but it would take a bit of thought to ensure the niches are unique enough to have each male look completely different from the other male while still having them keep the rock-paper-scissor dynamic and sharing the same rough habitat location (they are still competing for same females, so they must all live close to those females to compete for them)
I'd also mention that the female has to be able to produce the young of each male. if the males are different enough it becomes hard to believe that they are all capable of successful mating with the female. For that reason id suggest making all the males still have a fairly similar physiology. Perhaps the look different due to a comparatively superficial outward difference, most obviously their pelt/camouflage used to sneak up on their preferred prey, such that they can still be physically rather similar while still having very dynamic differences.
