You have an interesting premise, but there are some problems with it. Nothing that is impossible to explain or deal with, but some changes are probably necessary. I'll be trying to dissect what you've provided and highlight problems and possible solutions both.
There are almost no species where the majority of births survive to adulthood: the last two hundred years or so of low infant mortality rates in the developed world are a glaring exception. There's a reason why third-world countries (even now) are prone to families with six or more children: it's because, historically, that was what it took to average two surviving adult children (the minimum to maintain population levels), because less than half of newborns would survive their first two years. And this is with humans having great parental investment in each child: almost everything else in the animal kingdom has even worse mortality rates before adulthood.
As such, a high mortality rate in your species' young is absolutely normal; if even half of their offspring reach maturity (you indicate that medicine is basically nonexistent, and I'm not sure if your species is even sapient), they would have to breed more slowly than humans do and have huge investment in each child, which is incompatible with your premise that they seem prone to killing their own kind. Litters of one thus seem improbable; I would suggest an average of two or three to a litter, with a wider possible range of perhaps one to five.
If you want high mortality in the young to be enforced by adults killing the young, this probably entails infanticide; lions make a good real-world example of what you'd end up with. A male who kills another male and claims that male's mate for his own would be killing any young the female already had, so as to make sure more resources would go to the young he will eventually have.
Problems arise for you when you ask why the adults would be prone to killing each other: they might be battling for mates, but if so only one sex (this is almost always the males) will be doing the fighting. I don't see any other likely justifications for apex predators to actually fight their own species; evolution is ruthless in dealing with those who waste their energies on endeavors that don't help them reproduce. As such, their energies need to go mostly to hunting and finding/keeping mates, not killing their own kind and risking disabling injuries for the sake of...well, nothing (unless it's to acquire a mate, hence the exception for mating battles I mentioned). Apex predators tend to favor intimidation or posturing when dealing with their own kind, ways to establish dominance and superiority without the costs and risks of actively fighting (wherein both sides are likely to lose).
You also point out that reproduction periods end in bloodbaths; this implies that your species generally move in groups or packs like wolves, because solitary hunters would have a hard time coming together in significant numbers. Both sexes fighting it out to the last one of each standing is both absurd and wasteful: evolution would favor either only the males fighting to prove themselves to the females, or just settling choice of mates more peacefully. The second case fails to meet your objective, but having one adult male to every two or three females is entirely plausible if the battles are frequently fatal.