OK, here it is: a washer bottle filter! My design is a lot smaller and different than the original one, but it should serve the same purpose and it fits fine, is still easy to remove for cleaning, and allows plenty of flow for refilling bottle.
The best way I found to make a screen for the filter was to utilise Cura's internal support designs, so this is a two-piece thing that one must melt together (very easy with a soldering iron)
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https://www.thingiverse.com/thing:6919514
I know the assembly process of this is subideal, but the end product is quite sturdy and functional. If anybody could figure out a way to print this in once piece or make the screen without my sketchy support-based workaround.
Hope this is useful for people