I wanted to add something to the technical post I just wrote to the OP above...
I want you to know that most-all of the members of this forum, including myself, totally understand the terrible decision you have to make regarding this baby bird that you have probably been visiting for quite a while and have been planning on bringing into your family this upcoming week. I'm sure you've been quite excited about it for a long time, buying a cage, toys, food, etc. for him already...And like I mentioned in my first post, it's likely that if this bird continues to act this way, or is actually ill, that it will be hard to sell him, which the owner of the shop is well aware of too.
I'm hoping that you can sit down with the shop owner and ask him/her all of the questions about the baby's history, breeder, hand-raising/feeding history, about his/her siblings and their health and personalities/behavior in comparison to this baby's, and about the shop's policies regarding Live-Animal Sales, Vet Bills, etc. directly after purchase. Hopefully they will answer you honestly and without any hostility. If they obviously lie to you, seem like they are upset, angry, frustrated, etc. about answering any of these questions, or they are simply unwilling to answer any of them, then this is an indication that this pet shop is one to stay away from and never buy another live animal from again, as they only care about getting the birds out of their shop and collecting their money...which unfortunately is not uncommon. If it's a Petco, Petsmart, Petland, or other big-box pet shop, then they all have policies in-writing, which you sign on the day you take the live-animal home, to protect you, the animal, and the store, so that actually makes this process a lot easier for everyone, as they all cover the vet bills and/or refund you money in-full if the bird is found to be sick by a licensed Avian Vet within a certain number of days from the date of purchase.
***I forgot to ask you something, stupidly, as it's an important question:
-How long have you been going to visit this bird? Has this bird been doing this breathing thing from the very first time you went to see him/interacted with him, or did this start suddenly sometime AFTER you had already been visiting the bird?
-How is this bird's personality, and what is it's level of tameness? Did the bird step-up for you right away, does he allow you to touch him/pet him, etc.? Does he get upset when you try to touch him, does he squawk/scream at you or has he nipped or even bitten you when you're touching him or getting him to simply step-up?
***These are important questions to ask yourself, as obviously if you've been visiting this baby for a while and he/she only started exhibiting this breathing behavior some time after the first visit, or after many visits of acting normally, then that makes it quite obvious that the bird either contracted an infection or illness of some kind, OR as already mentioned by someone else may be living under a lamp or in a temperature that is far too high for him. IN CONTRAST, if this bird did this breathing behavior from the first time you visited him and EVERY TIME you have visited him since, AND the breathing behavior is the ONLY sign/symptom of illness that the baby is displaying, then it's likely that the bird is actually not sick, but rather extremely nervous, anxious, and worked-up. This would definitely be reinforced if the baby is also not very tame, gets upset and squawks/screams, nips, or bites whenever you have tried to touch him or get him to step-up...
Also, I wanted to say that there is one situation where I personally have seen this EXACT BEHAVIOR in one of my own birds, a female American/English hybrid Budgie that I hand-raised myself name Sid, and who is now almost 3 years old and is perfectly fine health-wise. Sid is one of 8 American/English hybrid Budgies that I bred in the last clutch of birds I ever bred, retiring all of my breeder-Budgies right after this clutch. I pulled all 8 of the babies at 2 weeks old and hand-raised/hand-fed them, keeping them in a brooder and not seeing their parents again. All 8 of them are extremely tame, easy to handle, fly to me, etc. There were 6 males and 2 females, and all of them behaved completely normally from day 1...except for Sid's breathing. Now Sid would step-up and allow me to pet her with no problems at all, but I noticed very early on, and I'm talking just weeks, maybe a month or so after she weaned, that she would just be sitting in her cage on a perch, not moving around, not playing, and she would be breathing just like the baby in your video, exactly like your baby. Whenever I would go over to her cage, which she shared with all of her 7 siblings at this point as they were not mature and could not breed, and it was a huge, massive flight cage by the way, she would come right over to me, and I'd get her out, she'd step-up, fly around, etc. But her breathing was so heavy and labored I thought she for sure had an Upper Respiratory Infection, which meant they all did, all 8 of them, and possibly my larger parrots in the house too...Uhg. So right to my Certified Avian Vet, who did Fecal Cultures, as well as Cultures from her nostrils/nares, and flushed her crop and took cultures from the flushings...And besides a bit of a bacterial infection showing in her Fecal samples, which was not anything that I treated medically but rather with Apple Cider Vinegar and Probiotics, everything else was negative. My CAV said that it could be a temperature thing, but she did it constantly, no matter where she was. Of course at the vet she did it, but that's not uncommon as they all have anxiety at the vet. But what I took note of is that her breathing didn't get any heavier at that vet, it didn't change, it was the same...This went on for a few months. She was totally fine other than the breathing. Well once the Budgie siblings hit around 7-8 months old I put the 2 females in their own cage and the 6 males in their own cage, but it was basically just a huge, separated flight cage, they were all still right next to each other.(I obviously didn't want any in-breeding, or breeding at all)...Sid's breathing seemed to go to normal now and then, which was a big improvement from it being all the time, but she still did it at least a couple of times a day. Finally I decided to give Sid her own cage, by herself, and I put it in my bedroom where there were no other birds during the day. I moved her and figured that she would naturally need time to adjust to the new cage, new toys, new room, etc. But when I came back into my bedroom to check on Sid about an hour or so later, her breathing was completely normal. And from that point on for the rest of the day it was normal, UNTIL bedtime, when my 4 larger parrots come into my bedroom to their sleep cages, which is where they sleep at night, every night. The larger birds didn't even notice Sid in her cage, but Sid certainly noticed them. Her breathing went crazy as soon as they came in the room, and even after I covered all 4 of them, they were silent, and I cover Sid, she continued to have really labored, heavy breathing all night. And this continued, when any bird came into the room Sid had the heavy, labored breathing, and that also included my dogs coming into my room, but when Sid was in her own cage in a room by herself or only with me, she was totally fine...
This continues to this day, and my CAV totally agrees at this point that it's an anxiety issue that Sid has being around other animals at all. He offered to medicate her for anxiety, but I didn't want her to be sedated all the time, falling off her perches, not able to play, etc. So Sid is still in her own cage, in my room by herself during the day, and at night she is covered and put to sleep about an hour or longer before the 4 larger birds come up to bed. This for the most part works.
I'm telling you this story because the breathing is EXACTLY the same as your little guy in the pet shop, and though I don't know any other information about the timing of this labored breathing, when it happens, what is going on when it happens, if he's around certain other birds or people, under heat lamps or just really bright shop lights, etc., I thought that I would relay this info to you anyway. I'm in no way saying that I'm sure this is the problem with this Green Cheek, but rather saying it's a possibility and you have to look at the WHOLE PICTURE of what is going on, who, when, where, etc. and then put it all together...And a CAV exam and cultures is going to be a must for this baby no matter what.
And btw, if the shop owner won't allow you to even pay out of your own pocket to take the baby to a CAV for an exam and cultures to find out if he/she is okay, then that's a horrible red-flag...