If it says the module is already installed, that's fine.
Do you know if you have the the bcma, ssb and b43 modules blacklisted? You can check by looking at the control panel -> system stats -> boot.
You can also check with iwconfig if you have eth1 (used by broadcom for wifi) and examine dmesg for details on broadcom, wifi or eth0.
The debian version of the wl driver is 6.30.223.248, whereas the tinycore version is 6.30.223.271, but the debian version says it supports "BCM4360-, and BCM4352-based hardware" - since the tinycore version appears to be more recent, you would think it also supports those?
tinycorelinux works fine on my mac mini