I guess I'm missing something. Why would you need a bucket to unload a dump bed? Isn't the purpose of a dump bed to be able to unload it by dumping?
Load it with your big bucket, use the ATV to move it where you want it, raise the bed and dump as you go to spread. Then use a pull behind rake on the ATV to further spread and level.
If you don’t care about spilling the material you are loading that would work fine. I want to scoop the material from my big dump trailer and take it directly to where it goes. The big trailer will be parked in an area that has crushed asphalt on the ground. I won’t be loading from a big pile where I don’t care about spillage.
I have a big dumpster that is five feet wide. Many times I have tried to clean up debris from various projects with my skid steer and dump it in the dumpster. It is amazing how much ends up on the ground due to the bucket being 2’ wider than the dumpster.