Higher risk of cavities is unfortunately a side effect of EDS and I’d first tell your dentist if you have this and ask if they know about EDS and teeth. Also iodine is an extremely common allergen and many places no longer recommend it for sterilization. If you do get recommended this or decide to try it for yourself (it’s otc?) please please please spot test it! A whole-mouth reaction would be terrible.
It is mainly been studied in children, but has shown to reduce the bacteria that causes dental carries which is the dental term for decay. The product marketed for dental use is called Povi-One, which is 10% povidone iodine. 10% povidone iodine is also readily available over-the-counter in pre-soaked swabs so they may permit you to even use that if they think it would be of any benefit.
I just wanna share it as something to ask about because it seems to be mainly used for children with a high risk of tooth decay, and your dentist may not have previously known about or considered whether it may help older individual individuals also experiencing dental issues and so it may be useful for you to specifically ask about it.