A couple of thoughts:

I second the Blue Lug recommendation. Great shop with great staff.

ride2rock.jp is the website for the Japanese distributor of what looks like most of the QBP lineup. You can check the dealers of various brands on there. That might help you find more stores that stock the bikes you're looking for.

Don't forget to check Rakuten and Amazon. Rakuten is all resellers who are all competing with each other on price so sometimes you can find crazy deals as someone needs to get rid of last year's stock. Of course, no test ride this way...

I also second the Tokyo Cycle Club recommendation. You can post a wanted ad there. Some of those guys have several bikes and might have a currently unridden fixed gear in the lineup.

There's a guy named Tim Smith who runs his own shop called GS Astuto (gsastuto.com). He's hooked into the track racing scene and can let you know about open days at the velodromes. He may also be able to help you find a bike, and will certainly let you do some test rides if he has something in stock. He might be able to help you find something nice but second hand. He's also just an awesome guy.

Hope some of that helps! Good luck in your search.