Paul Wells, the managing director at the track told me that for a bike to even be considered to be allowed on the track the bike must have 11.2 inches of bottom bracket clearance.
The Fuji bikes work, http://www.fujibikes.com/Specialty/Track/TrackComp.aspx
The problems so far in finding a bike that will work are...
1. Websites don't always publish this information
2. Many "track bikes" are urban fixes with low BB heights.
Known bikes that cannot be ridden on the track
- Surly Steamroller
- Redline 925
Which bikes do work? Please share this knowledge!