I have two of the filters like you have. Actually I have three, but one is in real rough shape. I soaked mine in a degreasing solvent tank at work, then a carb cleaning solvent tank, and finally a very strong soap. After all that soaking, I was still getting dry dirt pellets falling out. I finally pulled all the fibers out after removing the baffle on the bottom.
You are correct in your observations, these filters do not come apart. Most guy's cut the guts out and install a paper element inside and use the lid to seal the top of the element. I haven't decided how I am going to fix mine yet, but i do know, I wasn't going to get all that fiber material clean. It was full of dried on bits even after all the soaking I did to it. I believe the fibers are cocoa fibers like you would use for a planter from the garden center.

Joe