It looks like the request filtering is actually filtering for a blank file name. Therefore you have to add this to the request filtering block in the web.config:
<?xml version="1.0" encoding="UTF-8"?>
<configuration>
<system.webServer>
<security>
<requestFiltering>
<fileExtensions allowUnlisted="true">
<remove fileExtension="." />
<add fileExtension="." allowed="true" />
</fileExtensions>
</requestFiltering>
</security>
</system.webServer>
</configuration>
It's obvious now, but really I think its a massive gotcha.
More info: IIS 7 Not Serving Files - 404.7 Error