Here are two websites that might be helpful. The one from Iceland Exxcursions (re.is/iceland-on-your-own/) is valid until October 15th and Straeto (straeto.is/english/plan-your-journey/schedules-and-maps) goes all year round for most routes. The highland routes 610, 14 and 17 (Iceland Excursions) might be closed in September depending on the weather, the rest should be going.
Biking is a pain in Iceland, a windy and rainy country with lots of mountains and drivers who are not used to bikes.
Bring a sleeping bag for hostels (cheaper).