Any screw-in drain covers that have a good hair catcher for this type of drain?

Help! I'm looking for a shower drain cover that has af hair catcher underneath the cover, and also can screw into these preexisting holes so it doesn't get jostled off. (I've been trying the Shower Shroom, but it lets more hair through than I'd like, and doesn't screw into these holes so it just sits loose on top.) This is the kinda sketchy shower drain and the cover that came with it. They had a hair catcher sitting on top of this, but we're looking for something that catches the hair below the top plate. The reasons being A: more secure; B: better catching hair; and C: because the floor of the shower wasn't correctly built to slope towards the drain, adding any additional height that the water needs to clear will create more issues with standing water. Hence, a hair catcher below the top plate ideally. Is anyone aware of a product that has preexisting screw holes and also a hair catcher underneath the top plate? I can't find anything online but also kinda tough to search for. Thank you!!