Well that's embarrassing, I thought I was using the official joymouse but I double checked and it turns out it was a third party version. The official one works fine with both buttons. Unfortunately it displays a pointless dialog box at startup saying to "use exchange to control it", any way to disable that?
Check the tool types on the Joymouse program icon. We need (not bracketed):
CX_POPUP=NO
DONOTWAIT
This is assuming that you're using the Joymouse program icon to launch it or WBStartup, as opposed to the Run command. If Joymouse doesn't have an icon, then create a tool icon and enter those tool types into it and try that. You can have it launch in a script as if launched through clicking on its icon, but we won't go there unless you need to.