I believe you already know the answer.
You have 2 options, provide your own border (that's a lot of unnecessary work) or position an element with occludes the border (the problem with that is that you can only have a solid background color, but maybe that's fine).
To my knowledge there's not anything you can do to make this workout nicely in every browser out there. I like your style though, it's the right approach but probably not a problem you'll be able to solve in a satisfying manner.
My general opinion on these topics is that you should not try and do things with the web that requires A) excessive effort or B) a markup solution which is not entirely obvious to begin with. The web has limitations and you would do well to ad-her to them rather than trying to work around those limitations.
So I'm forced to ask, what's the problem with ?