The FB.init documentation: https://developers.facebook.com/docs/reference/javascript/FB.init/#flash gives you some options for doing something better than displaying a white background if your application will not work with wmode="opaque".
Adobe Flash applications on facebook.com
For Canvas applications using Adobe Flash, wmode="opaque" is preferred
whenever possible. We have found that, on modern browsers with
hardware compositing, there is generally no performance degradation to
using wmode="opaque". Otherwise, Facebook will, by default, hide your
Flash objects when popup events occur, and redisplay them when the
popup is dismissed.
If you need to use wmode="window", and would like to control this
behavior (such as also showing text or an image when this happens) you
can provide a function into the hideFlashCallback parameter to
FB.init. hideFlashCallback takes a state field as part of the passed
in parameters saying whether the window is being opened or closed.
This is the default implementation that you'll be overriding if you
provide one, but may also give you an idea of what your override would
look like:
function(params) {
var candidates = window.document.getElementsByTagName('object');
for (var i = 0; i < candidates.length; i++) {
var elem = candidates[i];
if (elem.type != "application/x-shockwave-flash") {
continue;
}
var good = false;
for (var j = 0; j < elem.childNodes.length; j++) {
if (elem.childNodes[j].nodeName == "PARAM" && elem.childNodes[j].name == "wmode") {
if (elem.childNodes[j].value != "window" && elem.childNodes[j].value != "default") {
good = true;
}
}
}
if (!good) {
if (params.state == 'opened') {
elem.style.old_visibility = elem.style.visibility;
elem.style.visibility = 'hidden';
} else if (params.state == 'closed') {
elem.style.visibility = elem.style.old_visibility;
elem.style.old_visibility = '';
}
}
}
}
Note: Some UI methods like stream.publish and stream.share can be used
without registering an app or calling this method. If you are using an
app id, all methods must be called after this method.