It's a good thing that your name check has been expedited. It's the only thing that can be done for now. You just have to make sure that as soon as the results came back, your application is adjudicated.
Have faith that it's going to work out now that the check was expedited !
I talked to a couple of lawyers myself and none of them mentioned a mandamus. With so little time left, they both advised me to talk to the senator's office (one of them admitting that he won't be able to do much).
You can find more info on these lawsuits here:
http://en.wikibooks.org/wiki/FBI_name_check
I believe it's going to work out without a lawsuit. You can call KCC and ask what is the crucial step that needs to be completed before September 30th - which AOS application processing step MUST be completed before the end of the month. After the 30th, there won't be any visa numbers available for DV-2008. I'm not sure when, during the AOS process, a visa number is given - probably at approval time. So, if you're assigned one before the deadline, other processing steps, for instance, the welcoming letter, GC, can be done after (you don't need to have the actual GC in your wallet on October 1st). ~
This is just MY thinking/opinion !
Be prepared for the worst and hope for the best !
The very best of luck !