Having done similar offset holes in wood and tile your first solution is very close to what I would try but with some adjustments to the plan
As always test the whole process first on scrap material
Do experiments with the forstner bit and I also suggest a normal hole saw with the 1/4 inch bit removed.
Try a bunch of things on scrap and see how it works and what you like and test the depth marking on your bit so you have a good feel for the depth control.
Find a piece of wood (not laminated) at least an inch thick to act as a bit guide, shipping pallets often use very hard dense woods great for this plan
Before you drill the truck bed
Score surface round with a knife to cut the wood grain to get a nice top edge on your hole.
Practice scoring during your experiment trials too.
Get a good feel for dragging the utility knife in a nice circle.
The offset washer should be rotated towards the outside walls of the box, the washer is trying to maximize the wood thickness exposed to the countersunk floor runner so that the wood isn’t thin and prone to breaking between the washer and the runner.
Your photo shows the washer isn’t rotated towards the wall as it would have been originally
Best of luck
-s
