In reference to the original question;
I think that you may have a valid point.
Before I had rhinoplasty, I used to sweat mildly. It was never an issue. The rhinoplasty didn't go to plan due to a knock I received while still wearing the splint.
This lead to a 2nd rhinoplasty which has now made the structure of my nose quite weak and has lead to breathing problems through my nose. I may have a slightly deviated septum on which I'm currently consulting doctors.
I am now a mouth breather and like yourself, I am in shape, exercise regulary and there is about 12% body fat on me, just like before I had the surgery.
Only now, as I can't take in large if any amounts of air through the nose, after purposefully trying to breath through my nose, I then turn to automatic mouth breathing. Like you, within minutes I begin to sweat and by the end of for example an hours 5 a side football, my shirt is drenched, front to back. I sweat profusley, others even notice it and can't believe how much I sweat, it even happens in night clubs if I'm dancing. The only difference being that I can now not breathe properly through my nose, so it must be that.
I'm told that when you breathe through your nose, that the turbinates warm up, clean and humidify the air as it passes through the nasal passage into the lungs. If you are breathing directly through your mouth then the air isn't being warmed up or humidified. The body must then counteract the cooler air by increasing the body temparature, that and with the already increased body temperature through exercising must cause the body to overheat, which then leads the body to sweat profusely in order to cool down. It seems to me like a vicious cycle and again tells me the importance of the nose and why you shouldn't mess with it if the breathing is already ok.
The nose is vital to everday comfort, breathing, smell, taste and mucus production that helps to keep the nose, throat and mouth moist as well as fight off infection.
I can honestly say that having had rhinoplasty over a slightly bent nose due to a sporting injury was the worst decision I have made as it now affects my life daily.
Hopefully the ENT specialist will be able to identify and sort the problem which I suspect is due to a deviated septum blocking the left nasal passage.