So I ended up pulling up the seat after all. You guys were right, there is definitely not enough room to work with before the coupling, and pulling up the seat was super easy. I will still try to flare the brake line after the coupling a bit further back from the spot marked by the red line. I really don't want to mess with the other parts because they don't look bad enough that they'll break, but I am afraid of stripping or tearing the brake lines or having to replace the rubber brake hoses if I screw something up. I can deal with that in the summer when I have more time. I ended up buying the ni-copp line (yes, I had a bit of sticker shock when they rung up the 25ft spool) and fittings at an auto parts store. Scratch that. 3 auto parts stores. I'm starting to remember why I hate going to auto parts stores versus just buying stuff online. Constantly over priced, constantly out of stock, and constantly wrong parts sold. I wish I planned this out better, but I digress.
It's amazing how much easier ni-copp is to work with. And now I have the fittings, brass couplings and the brake line. Hopefully I can finish all of this tomorrow. I am worried that I'll end up stripping or breaking the bleeder valves on the calipers though. Planning to blast them with WD-40 and hopefully that will be enough.
As for covering the brake lines with a protective shielding, would I just be able to spray them with rubberized undercoating to form that layer of protection? Or is that not a good idea.
P.S.: Numbchux, thank you very much for the detailed write up. I will definitely follow all of those steps when I get around to redoing this the proper way.