Heh, I've been watching this thread and hoping that someone else would figure this out because I didn't know the answer.
I did a little digging and came up with a few papers that are on point here:
Mayo, F. R. J. Am. Chem. Soc. 1937, 59, 1655-1657.
Mayo, F. R. Chem. Rev. 1940, 40, 351-412.
You might not like the answer, and there isn't a whole lot of experimental evidence, as Mayo admits. There is enough to suggest some steps though.
First off, the bromine ends up between the two carbonyls, everyone seems to agree on that. The next part is a little tricky though. Mayo did some experiments where he showed that this initial product doesn't rearrange in a vacuum, but it rearranges rapidly in the presence of air. His hypothesis is that the initally formed bromo-beta-ketoester is dehalogenated by either oxygen or hydrogen peroxide. That forms a radical between the two carbonyls which can then rearrange to the other alpha position and then react with a bromine atom carrying on the chain. (Scheme attached). The process may also be affected by the presence of small amounts of HBr, but the role of this reagent isn't clear, it might not even be necessary.
The argument for why you get the rearranged product has to do with the stability of the radical intermediates. The first on that you form is tertiary and therefore relatively low in energy. The rearranged radical is secondary and therefore higher in energy and more reactive. However, the two brominated compounds are relatively similar in energy, so the reaction is driven by the fact that it is harder to go from the product to the secondary radical than it is to go from the starting material to the tertiary radical.
This is specific to beta-ketoester compounds, the rearrangement of simple alpha-bromo-ketones is different. See: Tetrahedron Lett. 1978, 19, 1659-1662.
All in all very interesting! Definitely not obvious though.