I take it that the connector that is on the battery now will not fit into the Equinox Balancer end that says 3S?

I noticed that the battery you have does not have a balancing tap on it, but they sold you a balancer? I didn't think you could balance a battery through the discharge/charge leads alone (main black and red wires)?

My setup is a bit different, and all of my Thunder Power batteries have balancer taps, 4-5 smaller wires with a little black plug.

To me, it does not look like you are setup to charge that battery, it won't hook-up to that balancer plug.