ugh aren't they disgusting sometimes? I have had this problem with past dogs and my two would get to it, and have had when I haven't been quick enough. With my cat Becker, he sleeps in the laundry at night, and has his enclosure outside, and comes inside too. He does his poop normally at night, so in the morning when i let him out of the laundry in the morning I empty the tray, clean it and it gets put back down in it's normal spot on the back porch.
So due to that I rarely have problems. I know it's difficult, but if you can't block the tray off, all you can do is empty it as soon as you possible can after the cat has used it.
Sorry I can't be more of a help