get a telehandler with the highest lifting capacity when the boom is straight out

put the forks on it.........then weld up an attachment for the forks that consist of something that allows a LONG STRONG pipe or steel beam to poke straight out forward on that telehandler

from that pipe sticking straight out from the telehandler you weld pieces of chain hanging straight down that are strong enough to hold the slate

on the end of that chain you weld a piece of steel that is flat, thick, and hangs parallel to the ground/slats when the chain is hanging down

make that piece of steel on that chain wide enough so that it can fit through the slats.......BUT.......make it long enough so that when it is turned at a 90 degree angle you now have that piece of steel under the slats hanging from the pipe and all you do is raise up the pipe and pick up the slats

thinking about it you might need to just put a pipe on each fork and then have 2 chains per pipe/per slat so that you would have 4 chains per slat near each corner

if the slats are 4X16 then a 22' pipe should be able to hang over 4 at a time.......if you get good you might be able to pick up 4.........move forward over another 4............drop everything down..........jiggle the chains down lower through the lower level and pick up 8 at a time stacked 2 deep before you back out of the house and drop them down and go back for more......doing it like this you could also stack them high outside as well

if the chain gives you trouble controlling the steel tabs on the bottom you could slip a piece of pipe on the last 6" of chain and weld the tabs to it and then you have pipe turning the tabs instead of chain

I hate hard work so that is how I would do it

here is a pic too....the black is the forks.....blue is the pipe......red is the chain.....green is the tabs

as shown drive up......have the pipe (or more like pipes from each fork) hang out over the slats........dribble the chains and tabs through......turn then so they cant slip out......lift

if you stay in the 4' dimension you should be able to extend out over 4 or 5 at a time probably more with strong enough pipe......and like I said if you picked up a set.......rolled forward and lowered down and picked up another set or maybe 2 you could save trips in and out of the house 

I would look to take advantage of them being slats......I can tell you what my lazy ass would NOT do and that is get caught bending over and picking them up.....not a chance in hell.....not 540+ times when I know it is not a one time job