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.