I think slate is an awesome bottom!
Have you considered running it like an under gravel filter without the gravel? Leave the cracks between the pieces open then leave a 1/4" space under the slate. Then run a filter supply line to under the slate. This you have something to collect all the settled debris as it works its way under the slate.
Just a thought!
P.S. I laid the same slate in my entry area and on my stairs and love the look! I looks even better when wet! I can tell the top piece will have some beautiful natural purple and rust tones to it.