If you do want to go the programmed start route for 22W T9 circline fluorescent lamps, I think a programmed start ballast for 9 inch 18W PL-L lamps is a good place to start since I have researched that 9 inch 18W PL-L lamps have somewhat similar characteristics compared to 22W T9 circline lamps.
An example of a programmed start ballast that should work for your needs is a Fulham Racehorse RHA-226-C ballast as I do see that it is listed for a single 9 inch 18W PL-L lamp.
See here:
https://ebay.io/m/dCXQO2