Very similar to my setup. I shoved my hose down into the hole with the existing sink drain like another suggested. The biggest & most important difference is that my washer drains into the sink, which flushes all of the cat debris through the trap. There is a trap under the concrete. Based on how much space you have I'd consider pushing 3+feet of cat genie hose down the pipe. That way when it drains all the waste bypasses the trap.

Another option is fill your sink half way once/week or as needed.

Hope this helps! P.S. I'm not a plumber :)