Stable fertility rate
Assume that this kind of reproduction rate is sufficient for most environments on earth, allowing humans to maintain a stable number.
That doesn't work. The reason why our population grows is that the death rate drops before the birth rate does. If you reduce the natural birth rate to match the modern death rate, then we'd have died out back when our death rate was higher. Perhaps during the Black Death or after one of the larger wars.
Our old birth rate was a reasonable match for our natural death rate. What it looks like is that our ability to change from our natural death rate to our modern death rate happens before we switch from our natural birth rate to our modern birth rate. In both cases, the switch comes from interfering with natural processes.
You can't get a stable population by reducing the natural birth rate. There are reasons why it needs to be as high as it is now. To get a stable population, you need a feedback mechanism. Historically this has been starvation. If there are too many humans, enough starve to death to bring things back into balance. And of course the fertility rate drops when women are starved. Our current ability to create food has far outstripped our ability to handle waste. Perhaps you want a waste-based feedback rate--if what is important is a stable population now.
If you were willing to accept a more mystical solution, you could make reproduction require three parts: a mother, a father, and an extra piece. The most mystical version would be something like a soul. Until someone dies, there isn't a soul available to quicken a fetus. Fertility doesn't happen.
You could also make more physical versions, perhaps a gem-like apparatus that appears in a person's forehead. The mother might have to swallow a gem from a corpse to provide a framework for her ovum. The population size would be limited by the number of gems.
Estrous vs. menstrual cycles
I imagine that a civilized society would still have sex during other times of the year but we would not be able to conceive.
This is inconsistent with an estrous cycle, which is normally what being "in heat" means. In an estrous cycle, sexual desire is limited to times of fertility. What you are describing is just a menstrual cycle with a different cycle. Instead of a monthly cycle, you're suggesting more of an annual cycle.
This is much more like the current situation than anything we'd describe as "in-heat". If you want this, then you should stop using the term "in-heat". If you want to use the term "in-heat", you should understand what it means. And what it means is no sex outside of heat, whether natural or artificially induced.
If women could only get pregnant at one time of year, then birth control would be unnecessary for most of the year. Just don't have sex in June or whenever.
Fooling the cycle
How does the cycle know that it's time? Day/night cycle? Temperature? What? Whatever it may be, they'd explore ways to change the cycle. Much of the effort that we put into birth control might find its way into birth promotion instead.
I actually think that it would be more interesting to have a true in-heat scenario. People would only have sex when the female was in-heat. To have intercourse outside the normal cycle, the female would need to be artificially stimulated. Males would respond to the female arousal.
In-heat females would take time off from work and seclude themselves until normal pheromones resumed. Romantic getaways would involve really getting away. Only out-of-heat females would be able to make contact with the couple.
The concept of rape would change. They'd really blame the female, since it would be the female desire that drove sexual encounters. Perhaps they'd have laws against in-heat females exposing males to their pheromones during their in-heat period.