The hoop backs are steam bent beech, so DIY replacement will be all but impossible, however, you could try making up curved cauls to match the curvature of the back where the damage is, dripping a liquid epoxy resin (such as West System) into the cracks, then cramping with cast iron G-clamps and leaving for 24 hours.
The wooden cauls would need to be heavily waxed to ensure release and they will need to be a good fit to the curves. You will need proper cast iron G- cramps to achieve the clamping pressure required. The only glue with sufficient strength will be an epoxy
As @foxhole says, not an easy DIY repair. Or an easy pro repair for that matter