Heh heh... I've tried that too and it works well also. At the end, I settled on installing the bottle cap filter on the receiving bladder. It's faster pouring water through, then having the water go through the "push pull" cap. Also, with the filter on the receiving end, gunk washes away, rather than accumulating inside -- as it would in the photo above -- potentially slowing down water flow even more.
But either way, we are not talking about a huge world of difference. And either way, thumbs up for the UL mesh bottle cap prefilter. :)