Without knowing what CSS you tried to get this to work, I'll describe how I dealt with a similar problem.
I have a SWF that I want to fill the entire browser window, regardless of the window size. When the user resizes the window, the Event.RESIZE event fires in Flash and I update positioning accordingly.
However, the page also needs a navigation header that uses HTML. I added a 50-pixel high header div at the top of the window. This div is not inside the SWF div, so now the SWF div, with its height of 100%, extends 50 pixels past the bottom of the window. Without having to jump through all kinds of hoops with browser quirks, the easiest solution for me was overlay the header div over the SWF div, so I set the header div's position to absolute. I am not setting a z-index for either object -- position:absolute for the header div takes care of this.
Additionally, my header div contains a drop-down menu that overlaps the SWF when activated. This menu does not flicker during its animation and seems to work fine in all major browsers on Mac and Windows.
Also, I am not using SWFObject to embed the SWF. I'm using the Flash CS4 HTML publish code to make this work, so there may be an issue with SWFObject. Here is the embed code that Flash CS4 creates for me. You may be able to derive some settings here to use for your SWFObject settings (the AC_FL_RunContent javscript method is supplied in a separate JS file supplied by CS4 on publishing):
<script language="JavaScript" type="text/javascript">
AC_FL_RunContent(
'codebase', 'http://download.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=10,0,0,0',
'width', '100%',
'height', '100%',
'src', 'assets/flash/MySWF',
'quality', 'high',
'pluginspage', 'http://www.adobe.com/go/getflashplayer',
'align', 'middle',
'play', 'true',
'loop', 'true',
'scale', 'noscale',
'wmode', 'opaque',
'devicefont', 'false',
'id', 'MySWF',
'bgcolor', '#333333',
'name', 'MySWF',
'menu', 'true',
'allowFullScreen', 'true',
'allowScriptAccess', 'sameDomain',
'movie', 'assets/flash/MySWF',
'salign', ''
); //end AC code
</script>
<noscript>
<object classid="clsid:d27cdb6e-ae6d-11cf-96b8-444553540000" codebase="http://download.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=10,0,0,0" width="100%" height="100%" id="MySWF" align="middle">
<param name="allowScriptAccess" value="sameDomain" />
<param name="allowFullScreen" value="true" />
<param name="movie" value="assets/flash/MySWF.swf" /><param name="quality" value="high" /><param name="bgcolor" value="#333333" /><embed src="assets/flash/MySWF.swf" quality="high" bgcolor="#333333" width="100%" height="100%" name="MikeHoltzman" align="middle" allowScriptAccess="sameDomain" allowFullScreen="false" type="application/x-shockwave-flash" pluginspage="http://www.adobe.com/go/getflashplayer" />
</object>
</noscript>