You are getting married, starting a "new life together" and you think an immediate puppy of any breed is a good thing? Instead of relaxing and getting to know married life, you are bringing in something that is going to take a whole lot of work and time and attention. Getting a living pet is one huge event, getting married is a huge event... do them one at a time and enjoy it.
Getting a pet should be a separate event from anything else. It isn't like adding nuts to your ice cream sundae. It should BE the sundae. As suggested, take time to find people with them, go actually MEET Basenjis. What you read isn't the same as getting to know them. Or as Spock said on Star Trek... the having is not the same as the wanting. To even be searching for a puppy before you have met the breed up close and personal is a bit disconcerting. I am glad you know to go with only BCOA breeders, but this isn't a breed to get impulsively. Waiting a year isn't a "heart-breaker" ... it's hard, annoying, etc. but "heart-breaking"... no.
You are looking at something that will live 12 to 17 years, it's worth waiting. It's also worth doing more researching and getting out to meet them. An indication of lack of researching on the right places is that it just recently came to your attention that they generally have only one litter a year, typically fall or winter. That's one of the unique characteristics and I'd like to suggest you go to some better sources (like BCOA breeder's pages). Meet some dogs. Go to a great source, Basenji University, ( https://www.basenji.org/BasenjiU/niversityIndex.html ) and read more. Spend time here looking at behavior issues. I'm not suggesting your puppy will have those issues, but if you know ahead of time what problems can arise, you'll be ahead of the game in keeping them from developing with your puppy.
I am not, like others, going to dissuade you from wanting a puppy. If you are willing to do the work, read about positive training, use the time between now and the puppy to KNOW how to house train, crate train, basic obedience. A puppy is an exhausting but joyful thing. I love rescue and rescues, but wanting a puppy doesn't make someone selfish or bad. One suggestion I am almost reluctant to make is that you might still want to contact breeders and let them know if a puppy placement doesn't work out, you are willing to take an older puppy. Just please, meet some before you do. Because I promise you, a lot of the things you read may sound enchanting--- and you may find they are, or you may find it drives you up a wall. My first Basenji did more damage than the previous 25 years of Rottweilers and Chows. No exaggeration. Until you come in to find a dog climbing a bookcase to get your glasses, you don't appreciate the determination. Your definition of "puppy-proofing" will go to all new levels. And if you both didn't grow up with dogs, you're going to need amazing senses of humor.
As for BRAT, Camp Basenji or the few other rescues that are responsible, unlike 10 years ago, the number of young to even middle-aged and healthy Basenjis has dramatically dropped. It is great for the breed, harder for those looking for a younger healthy rescue. From hundreds, I have heard BRAT placed less than 100 last year. I remember when I first started looking and they had a few dozen available dogs. Right now, they have 4 listed.. a 4, 6, 11 and 12 yr old. The one in GA is closest to you, but not good with younger children should you plan to have some. Frankly, at this point if you want a young dog, much less a puppy, a responsible breeder is likely your best chance.
As for your question... since I've given you a boat load of unasked for advice... yes some Basenjis have puppies in spring/summer/early fall. Some even have 2 heat cycles a year. It's not the norm. And the reality is that most responsible breeders have homes already on a list, so while you MIGHT get one, at this point, most likely it would only be if there was enough people on the list dropping off.