I had a favorite pair of black cotton/spandex pants that fit beautifully, but had faded. I got a Rit "Back to Black" dye kit, and the result was terrific! They were restored to their former glory! However, the process was long and complicated, involving constant stirring over large boiling vat of dye, and then carefully removing them (making sure I didn't spill all over my kitchen's hardwood floor) and rinsing at length, and then transferring them to a boiling vat of fixative! It quite literally took hours of time.
I realize I'm not answering your question about "dry clean only" fabric, just saying the process was not a walk in the park. I think Rit suggests using "the hottest water safe for your fabric" and the longer you process the article, the more intense/better the coverage, I think. If you are gentle with the stirring (I stirred like I was one of the three Weird Sisters in Macbeth) maybe your jacket would be fine!
Q