There's a simpler solution. Just move the entire assembly almost 4 feet straight up and add track extensions to the bottom of the side rails. You will need longer cables or, you could just create a new attachment point 4 feet up the door.
You're still only moving the door 7 feet but most of it's travel will be vertical and little of it horizontal. I believe if you space the tracks 6" down from the ceiling everything should fit. The only cost in this proposal is the track extensions. In fact, since there's so little horizontal travel, those rails could be shortened enough to create new vertical rails. No cost retrofit.

I gained a lot of experience with garage door springs when I did my trailer. The springs can be very dangerous. Equal turns is what it's all about. Best left to a professional if you have no experience.