I know this is a delayed response, but I just wanted to follow up and thank everyone for their valuable input. Thanks to this thread, we have solved the issue.
What we did, is we installed the ball valve, which did help, but did not solve the issue completely. Since that was what my original post was about, I was grateful to receive that information.
An unknown that further complicated the situation, was that the pipe that both the sink and my pump line was feeding into, was QUITE clogged. Even though it had been snaked multiple times.
We ended up moving the pump input into the drain pipe further down the line (approx 2' - as far as we could go before other drains began to enter that line as well). This put the pump input AFTER the neighbor's drain input, and AFTER the clogged pipe as well. We ALSO tied into the vent between the pump input and the neighbors sink input as an extra precaution as we just needed to get the issue resolved.
Thank you to all who contributed. ALL of the information shared was useful for me.
