Agree with the others to try with the rental bikes to get a feel for the track and to try different sizes. Weight doesn’t really matter that much so something solid and with the right geometry wins out over latest tech layups. The sprint track bikes tend to hold up through crashes by adding another pound of carbon fiber to what would be in a road frame. Anything like a Felt TK2, Cervelo T1, Bianchi will not be the limiting factor until you get to a point that people are giving you bikes. Besides the bike you will eventually end up with a set of cogs and chainrings with the tools to change them so budget for that too.
If it is a 250m track you need to pay a bit more attention to the geometry with steeper angles higher bottom bracket and than on the bigger tracks where the road fixies work ok.