washer drain overflowing, tried cleaning vent
You're going to need to get an electric (or battery) powered handheld drum-type cable to do the job. Something similar to this;   https://www.cleaner.com/online_exclusives/2016/04/milwaukee_tool_introduces_cordless_drain_snake

Not saying you need to go out and buy one, but most rental centers have them.  With that you should be able to run through the trap with no problems.  And if needed, you can access the clean-out (like in the picture) in your main drain and run the cable through it. 

I've been doing building maintenance for 35 years now at a local hospital, and for what it's worth, I'd stay away from Draino and similar products. 

Hope this helps.