Fixed bike recommendations for track racing

ttcycle, the only "track" bike suitable for you in your price range, is the Fuji Track, but its smallest size is 49cm, whereas you may be closer to a 47cm.

There is a really good alterantive, already mentioned above, and that is the Giant Omnium. Built up from Stonehenge Cycles, at £600, its £100 above your £500 limit. I'm hoping its possible that you can somehow stretch to that, as the smallest size is a 44cm! The sizes for Giants are unusual, and a 44cm is probably closer to a 47-49cm, but I'm sure it would fit.

The Giants have compact geometry, sloping top tubes which means the seat tube measurement is smaller for a given "size"

You and lots of other people on this forum still do not understand that the important measurement for a bike is the effective top tube length. The seat tube has a bearing on standover, yes, but fuck up the TT and you'll be stuck with looking at stems at one end of the available size range and very probably issues with your saddle fore and aft postition.