My child spikes fever only at night

My son who will be four in Semptember was born with a very low core body temperature...34.5 Celsius. It took over six hours of the heating bed to bring his temperature up to 36.6.

When he was two months old he was prone to spiking fevers only during the night( between 11:00 p.m. and 2:00 a.m.) to between 39-40 degrees, but wouldn't show any other signs of ailment other than lethargy and discomfort due to fever and his temperature would be back to normal by mid morning.

Our family doctor had run blood tests on both him and me (I nursed) when he was 4 months and deemed him in perfect health but suspects I may have a thyroid problem and claimed his fevers are due to a problem with his body maintaining body temperature and suggested I reduce his clothing at night and keep his room on the cooler side, as well as, weaning him on to formula and food. I have done as my doctor has requested and all has been well until the past 5 months.

Every two-three weeks when I wake my son up to go to the bathroom, I find him burning with fever again. Every time I take him to the doctor, he claims it's just a virus, but when I talk to his pre-school about this, they say he is the only child with symptoms. His pre-school thinks it could be from teething, but my doctor claims teething doesn't cause fever.

I also have a three year old that doesn't show any ailments during the two week period and beyond of my son's fever.
My mother is type 2 diabetic, both her and I have boarder-line hyperthyroid, my son has autism (ASD) and was diagnosed with a premature lower stomach sphincter at two months of age.(which he has out grown)

My son has just woken from his nap and again his temperature is up (38.76 reading under the arm) I looked in his mouth and his gums seem raised and on the whitish side behind his last molars. Can his fevers be due to teething?