I suspect that you have left the quilt unfinished for 6+ years because you know deep down that the black fabric needs to be replaced. It is a lot of work and it will take some time but sometimes the effort is worth it. There really is no point in making and giving a quilt that can not be loved on and used like any other quilt.
I obviously vote to replace the black fabric. Just because you put some black batting or fusible behind it is not going to change the fact that this fabric was too thin to begin with and will not hold up any longer just because you can't see through it. Besides the thought of replacing all later is going to be so much more involved and difficult. I'd avoid that at all cost. Good Luck. It is a very lovely quilt.