Your species would likely have an easier time dealing with overpopulation issues than humanity
Humans are unusual among animals in that they have a combination of two physiological traits which by themselves already are uncommon among animals.
- Concealed ovulation: Human females do not produce any visible signals of high fertility or ovulation. In most animals, females produce very visible signs that they are close to ovulating to advertise to males that they are the best choice of mate because they are most likely to bear offspring. In extreme cases you have things such as the sexual swellings of baboons and many primates, but the ovulation of most species can be easily detected by things such as the scent of certain hormones in the urine. Humans don't, and this has been suggested to be a feature, not a bug, because it encourages males to stay around females and assist in the care of offspring (which increases survivorship in such a K-selected species as humans). For all the studies of humans suggesting that exposure to pheromones put out during ovulation might subconsciously affect human behavior, there is no evidence that humans can consciously detect if another human is ovulating and modify their behavior accordingly.
- Lack of a constrained mating season: Unlike most animals, humans do not have a strict timetable for breeding. Humans may be more likely to breed at certain periods of time, but our species has sex throughout the year as can be evidenced by the fact that birthdays span all 12 months. So it is not like other species where desire to mate peaks in a set period during the year and then dwindles to nothing later.
The combination of these two features makes human reproduction in general a Russian roulette among animals. Whenever humans mate, we have no way of knowing whether it will result in pregnancy or not because there is no set season when humans are fertile and there is no way of knowing whether the female is at peak fertility. Women often track their ovulation cycles to avoid peak fertility, but the fact they have to chart it and can't just tell they are fertile demonstrates how hard it is to keep track of these things.
This is how you get things like teen pregnancies, unplanned babies, and a huge market for contraceptives in humans. Our breeding system is extremely chaotic and hard to control or plan for. If human beings had a visible estrus cycle or a set breeding season population control would be exceedingly simple, just don't have sex during the extremely forecasted breeding season. Even if the females had a hard time controlling a desire to mate during the breeding season, males would be able to easily determine whether or not they wanted offspring by taking one look at the female and going "this female is in heat. I do not want/cannot support a child right now, so I'll just wait the few weeks until she is out of heat and there is no risk of producing a child". If human reproduction were as predictable as your species we wouldn't be in an overpopulation crisis right now.
If you want there to be an overpopulation crisis but keep the distinct heat season, the best way to do it is to have severe negative side effect for not reproducing during the breeding season. This exists in nature. Female ferrets will get very sick if they do not mate during the breeding season, and this cannot be cured except by having sex. The rump of baboons swells during estrous, and there is evidence that in addition to individual genetics the degree of swelling is correlated to how long they have gone without mating. If they don't mate the swellings get worse and worse until they negatively affect the health of the female.
Of course, the question then becomes why doesn't the species, if they are sapient, just invent methods of contraception early given how big of an issue controlling reproduction is for their society. To go back to the ferret example, it is the act of mating that resolves the sickness, so male ferrets with vasectomies work just fine. Things like this (or less extreme forms of barrier or hormonal contraceptives) would probably work in other species with similar mating habits. This is the kind of species that would invent the morning-after pill before the printing press.