Up sh*t creek...

"The one behind the washer and drier is the larger one at the base of the vent stack."

I would bet this is where you need to get in. You need in a bigger pipe than a sink drain.

That would make sense, because this is the spot where my grandma used to have rats trying to crawl into the house from. That tells me that there is a direct line to the city main from this opening. Not sure why I didn't think about this before. Possibly because this clean out is behind the washing machine and not easily accessible.