We have family property up in the mountains in Kyrenia, North Cyprus is an amazing place. A bit qwerky but I love it there. In early September it’ll still be 30 deg plus weather wise, so it’ll be really hot still. I’ve not done much riding there to be honest, but the military areas are pretty well defined.
Have a look at these guys, http://www.cyprusactive.com/about_us.html they might be able to help with bike hire, routes & riding. They are friends of friends of the in laws. They are Brits and have been running it for a while so might have some good info for you & be able to help with bike hire. You could ride in the mountains for weeks though & never see anyone else!!
Kyrenia is a good place & the harbour is really nice, there’s turtle beach, bellapais abbey, St Hilarion castle & depending when you’re there, the blue house, but you need your passports & to go onto military property for that but all are great places to see & go. There are great restaurants all over & the food is really amazing. Just watch out for the crazy local drivers!!!