The automatic resizing is working properly. However, sidebar column (gk-right) has a fixed width of "26%". Because of this, even though the div inside the sidebar is growing, the sidebar itself is not.
There's really no way to fix this within the JFBCInvite module. The "Confirmation" dialog that Facebook pops up can not be changed, so it needs to be that wide. Unless your sidebar can handle a variable width module in it, JFBCInvite will not work there.
Hope this is clear, and let us know if we can help in some other way. I know this isn't a great answer.