I always hire from within (when I can) because my theory is that if I don't promote my best and brightest, I risk losing them to my competitors. It also keeps my floor fresh and it's great for morale as we run a meritocracy around here. I also have a succession plan in place in case I lose one of my finance or sales managers. Then there is this - I was denied the opportunity to advance when I was a salesrep (in 1988) and it stuck with me.
Now, with that said, I recently promoted my best salesrperson, and she couldn't make the transition. I lost her, but was fortunate enough to bring her back in a different capacity (internet manager). I blame myself but in the process I also learned a valuable lesson. Perhaps it would have been best had I had her very well trained, and then I should have had her start to settle her own deals. I also told her that her best fit would have been as an internet manager, but she really wanted a shot at F&I. In retrospect, she had several challenges, among them that she had a difficult time learning Docupad.
Promotions don't always translate to success. I've seen successful salesreps fail at management, but I've never seen an unsuccessful salesrep make the transition, so that's that.
You've been at your store for 17 years, so let's start there. You should have a very good clientele base and therefore you should be making a good living. I think the best approach is to speak to your GM. Believe me, he doesn't want to lose you. He doesn't want you making other people money. You said that he only hires "experienced producers." I'm sure he's hit on some, but I am also certain that he's also hired some who have failed miserably. Remind him of that. Ask him if you can train on your own time, when you're off shift. Get him to think win/win. Tell him once you're trained, to let you settle your own deals, and then once you're successful doing that, you can move into a full time finance manager.
If he's smart and forward thinking, he will listen to you and work with you. If he's not and you really want to be a finance manager, find a store that will give you that opportunity. Being a finance manager is all about being able to quickly establish rapport with customers, and being a good closer. If you are both of those things, then by all means go ahead and bet on yourself.