$500.00 may not be enough for the time it's going to take you...that aside:
You don't want to remove the backing and batting if the top is in bad condition...not until you have secured the top in some way. What I've seen others do with a quilt like this is put a layer of tule over the top of the entire quilt. Get one that won't show a lot. Dark fabrics, get a dark tule, light fabrics, get light.
What you are doing is securing the top as is, no repairs.
I watched this done on a longarm (video online) and they put a new backing on the machine...laid the quilt (batting, old backing and all) over the new backing, laid the tule over that, then secured and quilted a large meander all over the quilt. That I would do for 300.00...I'd also add a thin layer of batting if needed.
If they want the top repaired for use...I'd find like colors (but use cotton or polished cottons), gently take the quilt apart, replace any blocks that can't be fixed (lot of hand stitching here)...once all seams and squares are secure..add a new backing/batting and machine quilt...I'd use Stich in the ditch or an allover pattern. If they want it to look tied...add some ties after all the quilting is finished.