Not exactly what I expected. That's why the photo helps. When I used a bucket I built a frame and had the bucket at a 45 degree angle with the top open. But that was inside where rain could not get in. Another popular configuration is the bucket laying flat with either the lid split to hold the bedding in or a board across the bottom half.
It sounds like you do not let them hatch in there, it's purely for egg laying. I had a hatch issue with mine but that is not on the table with yours.
I've had full sized fowl hens larger than your Games use a nest with a 6" high opening. I can't tell how high that opening is. This kind of problem is why that opening was only 6", the hens were scratching the bedding out so I raised the lower lip. That solved that problem. I cut tall grass from places I don't weed eat or mow and dry it. My bedding is close to hay but the stems are probably a little lighter than what you use.
I'm surprised that 6" isn't enough to keep the bedding in. The way I read this thread it's worked for years and it's only one hen that is scratching the bedding out? You could try cutting a smaller hole higher up on a bucket but I'd give her a different type of nest unless you are aggravated enough to eat her. The bedding is not the problem. Since it has worked for years the design is not the problem. The one hen is the problem.