A larger washer is in diameter, not thickness. Cork material can be purchased at most auto parts stores. It is gasket material. Cutting out the material can be fairly easily accomplished by using various diameter tubing. I use copper tubing, one for the diameter of the spindle and one for the outside diameter of the drag washer. Filing a sharp angle on the cutting end of the tubing insures a smooth, accurate cut. The thickness of the gasket material may vary. If too thick, then sand it to the proper thickness. Use the original washer as a template.
Good luck and let me know the results. Where in the Keys are you located? I spend Jan-Mar at Islamorada.