Millpond:
I start with a warning that I am no expert and that my last encounter with pilot operated check valves in a control valve (as opposed to on a cylinder) was many, many years ago and it involved manufacturing machinery. However, what I remember seems to make sense in light of the problems you are having. So I offer the following advice to use at your own risk.
Note that none of this involves the quick disconnects at the remote valve on your tractor (probably just behind the seat) where the hoses to the grapple connect. The only disconnects you need to work with are those at the front of the FEL where the long hoses from the rear remote connect to the short hoses to the grapple cylinder.
If I had your problem, here is what I would do and, first, why I would do it.
The nature of a directional valve with pilot operated check valves in both working ports (what I think you have) causes a residual pressure to remain in both hydraulic lines whenever the valve is returned to neutral while the cylinder is in motion.
The amount of residual pressure depends on the design of the mechanism for opening the check valves in the return line. Some require only a minimal hydraulic pressure to overcome a spring and the hydraulic pressure does not vary with the load. Others require hydraulic pressure that is proportional to the pressure in the return line, and in that case the residual pressure increases with increasing load on the cylinder. I know that is confusing, and very incomplete, but perhaps it is enough to understand why I am recommending the actions below.
Note that I said there will always be residual pressure in the lines when the cylinder is stopped by the valve while the cylinder is moving. But if the cylinder stops of its own accord (because it reaches the end of its stroke or the mechanism it is connected to stops moving) while the control valve is open, there should not be residual pressure in the return line. You can use that characteristic to release the pressure in the lines one line at a time.
Here is how:
1. Start the engine and let it idle at the lowest rpm possible (probably 800-1200 rpm, depending on the tractor).
2. Move the valve handle (slide) forward or backward as required to close the grapple completely (ie, until the claw stops moving), then release the handle. Now look at the grapple. One of two things will be true: One, the grapple cylinder will be extended fully, which means that the rod will have come out of the cylinder as far as it will go, and the cylinder will extend no further because the piston is against the gland; if this is the case go to step 3. Or Two, if the cylinder is not extended fully then the upper claw stopped moving, it will be because the claw struck something (the bottom lip if the bucket, the lower claw, etc.; If this is the case, just be sure that whatever the upper claw is stopped against will not be damaged if you put a bit more hydraulic pressure on the grapple. It should not be a problem because grapples are usually designed to close until the upper claw strikes something (the object being held or the bucket, etc.) and then stops moving with full hydraulic pressure behind it. But if you are concerned open the claw back up a bit and put a piece of wood or something similar between the upper claw and whatever it is striking. Then close the upper claw back down against the wood. Then go to step 3.
3. At this point the engine is idling and the grapple is as far closed as it will go because either the cylinder is fully extended or the upper claw is resting against a fixed object (wood, lower claw, etc.) You are now ready, as described in step 4, to momentarily put full hydraulic pressure on the cylinder in the direction that would make the grapple close further if it could. When you put hydraulic pressure on the cylinder, but it cannot move, that pressure build up in the control valve should open the check valve in the return circuit without at the same time increasing the pressure in the return circuit (because the cylinder can move no further, and therefor cannot force hydraulic fluid out the return line). Then, when you close the valve the return circuit should have no residual pressure. So here is what to do.
4. Push the control lever on the control valve all the way in the same direction you did before to close the grapple. Just push it all the way and pull it back with a smart motion. The lever does not have to be (probably should not be) moved as fast as you can or as slow as you can either. Your action will open the valve and direct hydraulic fluid from the pump to the base end of the cylinder (the end away from the rod), which will attempt to make the cylinder extend further. Of course the cylinder will not extend, but the build up of pressure in the path to the base end of the cylinder should open the pilot operated check valve that is in the path to the rod end of the cylinder. The hydraulic pressure in the circuit from the work port to the base end of the cylinder is the "pilot" that operates the check valve in the return work port from the rod end of the cylinder. And when the check valve in the return work port opens it will provide an open path to the tank (reservoir) for the hydraulic fluid in the return path from the rod end of the cylinder. When the return path is open to tank the residual pressure in the rod end hose will be relieved.
5. Since opening the valve diverts hydraulic fluid to the base end of the cylinder, and since the base end of the cylinder can not move any further, the pressure in the pump circuit will rise to the relief setting (probably 2,000 to 2500 psi). You will probably hear the engine slow a bit as the hydraulic pump loads up and then you will hear the buzzing of hydraulic fluid flowing through the relief valve. Since you have a new system (I understand), it should be properly protected by a system and/or control valve relief valve and no harm should come.
6. If all that works, you should now be able to disconnect the coupler that connects the short hose from rod end of the grapple cylinder to the longer hose that runs from the grapple back to the rear remote on the tractor. The coupler to disconnect should be mounted somewhere on or near the grapple.
7. Once the hose from the rod end of the cylinder is disconnected, the base end of the grapple cylinder will still be connected to the control valve by its separate hose. And that hose will probably have quite a bit of residual pressure in it. You bleed off that residual pressure in the same way you bled off the residual pressure in the rod end circuit hose, but the directions are reversed. Step 8 explains it.
8. Now, with the grapple closed, the rod end hose disconnected, and the base end hose connected to the cylinder, move the valve control handle in the opposite direction you did to close the grapple. Do it the same way: not real fast or real slow. Of course, the grapple will not open because the hydraulic fluid being directed by the valve to the rod end of the cylinder can not get there: the rod end hose has been disconnected. But that is okay, because the pressure build up in the rod end circuit in the valve will open the check valve in the base end circuit in the valve and allow fluid in the base end of the cylinder to bleed back into the tank through the open check valve. That relieves the pressure in the base hose and you should be able to disconnect it. So disconnect the coupler in the line coming from the base end of the cylinder at the grapple. If the pressure has been released it should disconnect easily.
9. If (a big IF) all that works, you now have disconnected both hoses from the grapple, and the cylinder will have no residual pressure. One hose going from the cylinder back to the valve will have residual pressure: the rod end hose, because step 8 above pressurized the rod end circuit in order to relieve residual pressure in the base end circuit. But the other hose will have no residual pressure: the base end hose. So to reconnect, first connect the base end hose from the control valve to the base end hose at the cylinder. Neither should have any residual pressure. Then repeat step to 4 (move the control valve handle in the direction to close the grapple, momentarily) to put pressure in the base end circuit. That should open the check valve in the rod end circuit, thereby relieving the residual pressure in that circuit. Then you should be able to connect the remaining lines.
10. If this does not work, or if it is too much hassle, then the suggestion by J_J of new "works under pressure" couplers is the best solution.
11. Good luck.