I'm pretty sure it can be fixed by changing the refreshSidebar function in basics.js to not use ajax to update the content of the charpane, but simply do a top.charpane.location.reload() instead. When I wrote the gCLI integration for ActiveChat a long time ago I copied some code from basics.js and cli.html, and ran into similar problems. Resolved it by bypassing refreshSidebar and just doing a normal refresh of the charpane.
As far as I recall GM scripts also doesn't trigger when using ajax to update the charpane, so maybe it's worth having another look at it.
In any case, I've re-jigged my layout a bit to at least look semi-decent when the content is updated via ajax, so hopefully it won't be as much of an issue after I upload the next version.