Does ENT perform rhinoplasty or should I ask for a referral?

Most general Ear Nose Throat doctors don't perform rhinoplasty. It is very common to have both cosmetic rhinoplasty done at the same time as nasal breathing improvement surgery like you're looking to do.
You could find a facial plastic surgeon who is board certified in both ENT and facial plastic surgery who would be able to treat both issues.

If you want to correct function while improving the appearance of your nose, a board-certified facial plastic surgeon is likely your best option. These specialists have typically had ENT training, as well as extensive training and experience in cosmetic procedures. Rhinoplasty is a complex surgical procedure that requires expertise, precision, and artistry to perform well. When you choose a board-certified facial plastic surgeon, you will get these qualities as well as an intricate knowledge of nasal anatomy to improve function as well.

Both ENT and plastic surgeons perform rhinoplasty. I tend to send the functional rhinoplasties to my ENT colleagues as they can get back into the sinuses if needed. If I were you I would likely seen the ENT as your problem sounds functional.