No, Non-Owners insurance, with or without an SR22 filing, As the name implies indicates that you do not own or regularly operate a vehicle. Non-Owners coverage is only secondary coverage to any insurance already on the vehicle being operated.
You can not have Non-Owners Insurance on a car nor a motorcycle because to buy the non-owners policy you must declare that you own no vehicle nor do you regulary operate an uninsured vehicle. If you own or regularly drive a vehicle, You must obtain appropriate liability coverage on that vehicle whether you own it or not. Non-Owners Insurance only covers you for un-planned, un-expected vehicle use. It will not cover you in any vehicle belonging to household member nor in any vehicle to which you have ready access.