Renting a car in Kyrgyzstan is a great idea if you don't want to be tied to minibus schedules. I would recommend renting a car directly from the airport: this gives you flexibility when driving around Issyk-Kul and the mountain roads. If you decide to rent a car in Kyrgyzstan through specialized websites, here's a useful resource for comparing prices and models
https://getrentacar.com/en-US/kyrgyzstan