Hi evan,
Yes, the B2 MUST be connected to the same network (SSID) as the Sonos speakers for the Brennan/Sonos system to work. The Brennan is passive as regards connecting to your home WiFi - it asks what you connect it to for an IP address - it is the job of your home WiFi system to supply this IP number and the Brennan just accepts whatever it is given.
Home WiFi/Ethernet addresses look like this IP=AAA.BBB.CCC.DDD generically - where each A,B,C and D represent a number in the range 0-9.
In a usual setup the first two are IP=192.168.CCC.DDD and for the Brennan/Sonos system to work the third number (say IP=192.160.1.DDD) must be the same, while the last numbers (DDD) MUST be different for the Brennan and each Sonos speaker.
From what you say you have some sort of mesh network/router system in your house - these are a bit of a black art to configure. I must warn you that the devices you want to use to connect to the Brennan's Web User Interface MUST also be on the same AAA.BBB.CCC part of your system, or they will not be able to see the Brennan.
Fred