The thing is, a new screen printing will never match the original 100%. So with the shadow of the original screen print still on the fabric, it'll look odd as you will never get the new screen to line up perfectly with the old print. In the case of the bag fabric being in very good shape, I would say repainting it by hand is a good option. Screen printing emulsion (paint) is available on amazon and ebay. You might want to practice on a piece of scrap fabric first, and if I'm not mistaken, acetone will clean any mistakes. The paint needs to be cured with heat, so a blow dryer or halogen light will do.
Otherwise, put the bag on a flatbed scanner and copy the print design in black and white (no gray), black being the painted areas. Measure a reference point on the bag (a line, perhaps) and ensure the image size is appropriately large (in inches), and that the design matches the actual size as closely as possible. Then take that image to a T-shirt screen printing guy.