Hi,
I don't have any expirience of this issue, but it sounds like it could be really hard to debug
One solution I can think of is to use a combination of iframe and location connection. The idea would be to embed a small external interface "controller" swf via a small iframe on the page which will listen to events from HTML via external interface and forward them on to your flex app via local connection. Reverse wise localconnection from app > controller > HTML.
I am pretty sure that local connection is unaffected by visibility of the div etc.
Check 1st that it is not a swfObject issue by mocking up your page with the standard <object> / <embed> method. If it is still broken , then try the above.
Let me know how you go...