Yes, you can take a bike out of T for up to 4 weeks.
If you don't plan on returning with it then the time limit doesn't matter.
Before applying for a carnet you need an official translation of the rego (aka Green Book) from the motor registration office, as the plate's Thai characters and the rego itself can only be read by Thai and Lao readers.
You can't get a carnet in Thailand itself. Try the CAA, failing that the Swiss club mentioned in another post.
Good luck, you will need it. But then, you are already on plan B, aren't you?