My daughter had a lot of ear infections and went through 3 sets of ear tubes when she was little, as well as adenoid surgery. We were given the option of having her tonsils taken out when she was three (along with ear tube and adenoid surgery), but we opted out at that time, since she didn't seem to be having any issues with them. How little we knew at the time...
Fast forward to when she was 7 years old and we went in to have her tonsils out (the reason was that she was having system-wide strep infections and severe strep/tonsil infections). The Pediatric ENT who performed the surgery (her ENT doctor was gone that day) came out of the surgery looking like he'd seen a ghost, he was so white. Not only were her tonsils overly large--they went way back into her throat, instead of hanging down, where you could see them--but when pathology/the lab opened them up to check them, there were COMPLETELY full of strep infection. And she had just had a lab test done before the surgery that came back stating she was free from strep!!
My point is, if your health insurance allows you to see a doctor or a specialist without prior approval from your doctor (many do), I'd make an appt with a Pediatric ENT and take my daughter there ASAP. If you have to have a referral, then I'd look for a new Pediatrician who will listen to you and give you the referral you need to take your daughter to a Pediatric ENT. If it's making her snore and affecting her sleep (it could be large adenoids as well), then by all means, please get her in so you can get her the help she needs.
BTW, now my daughter is almost 11 years old and hasn't had a cold or the flu or any other type of an infection in four years. The surgery was the best thing we've ever done.