If you don't mind adding some extra markup , there seems to be an easy solution.
You'll just have to use two div, one for each overflow.
For example:
<div class="outer">
<div class="middle">
<!-- your content here -->
</div>
</div>
And the following markup:
.outer {
overflow-y: visible;
}
.middle{
overflow-x: hidden;
}
Seems to do the job.
A little example here.