I think it would be possible to use tables, but at the cost of having to use trickery to get around the “breakage”. It would be extremely painful to implement.
Tables can only go on a work page after the {{fte:imslppage}} declaration has finished, which relegates them to the very bottom of a work page. For a couple of examples of pages with tables, see:
Well, this is a bug. It is not a bug of the FTE template, but for all Mediawiki parser functions, so it will probably have to wait until the upgrade so that we can check if the Mediawiki team has already fixed it.