I have a large shipment of kayak building supplies arriving this week and among them is “wood flour” - basically hyper-fine sawdust used to thicken epoxy for structural filling. The resulting mixture is about as dark as the darkest portion of that door and is sandable.
I can’t spare any epoxy but I’d be happy to share some wood flour. Unfortunately I can’t help with the actual work but don’t mind donating some filler to the cause.
What I’d suggest is pulling the door off the hinges and laying it flat. Attach a piece of wood wrapped in plastic wrap along the edge of the door to act as a dam. Fill the void with epoxy thickened to mustard consistency. It should be mostly self-leveling. Prep, sand and paint the entire door.