We have recently had an issue with water from our kitchen sink and our washing machine drain backing up and not draining properly. I tried in very non-artistic fashion to draw out the combination of plumbing I believe is involved. Both the sink and washing machine drain into piping along the same exterior wall and there is a cleanout in the exterior wall directly opposite the sink drain. I believe there is a vent drain going directly up from where the sink drain and cleanout meet, and I think the washing machine drain comes in from the side there as well. I have tried pouring draino into the cleanout twice, and it had no effect. So I went out today trying to use a drain snake through the cleanout, and was unable to get anything to go down the outlet to the sanitary. I got a flash light and looked carefully, and it appears there is a small ball, rock, or maybe something like an avocado pit stuck in the drain pipe.
First question, is there any reason there would be a ball check valve back in there for any reason? Backflow prevention from somewhere that broke loose maybe?
Second questions. Any ideas on how to get the object out without completely dismantling the wall/plumbing to get to it? My thought was to take a long drill bit and try to drill into it from the outside cleanout and then try to lift it up and out. Any thoughts on how that plan could go terribly wrong?
TIA for any help and some #fineassfriday for reading to the bottom before laughing at me.

https://instagr.am/p/CQ3r0yhl4re
First question, is there any reason there would be a ball check valve back in there for any reason? Backflow prevention from somewhere that broke loose maybe?
Second questions. Any ideas on how to get the object out without completely dismantling the wall/plumbing to get to it? My thought was to take a long drill bit and try to drill into it from the outside cleanout and then try to lift it up and out. Any thoughts on how that plan could go terribly wrong?
TIA for any help and some #fineassfriday for reading to the bottom before laughing at me.

https://instagr.am/p/CQ3r0yhl4re