It's all going to come off eventually. There are few ways to speed up the process. I guess just keep peeling is the way to go. You could try screening off the loose finish and then recoating after proper preparation, but you'd have to be sure to get it all off and down to a consistent surface prior to doing anything else.

You may have to resort to a resand.

I am not familiar with screening. What is involved in that process?

Professionals sometimes use a buffer and a 'screen' to abrade a previously finished floor to prepare it for coating. The screen looks like screening with grit particles in the construction. Hence, the term screening. It is a less aggressive sanding media and is commonly used in preparing a floor for recoating.

Some floors don't need screening at all before recoating, but rather receive a chemical cleaning instead.