A real track bike will beat you up on the roads. The default design in a track frame is 'sprinterish.' Most track events are under one hour, and most tracks are smoother than what your local highway department puts down.
Also, a typical track bike will have a short front center, which can lead to an excess of weight transfer under braking on the road. IOW, endo! On the track, it's no trouble at all due to the lack of brakes.
If you're set on fixed gear on the road, convert a road frame, preferably a soft-riding unit, as you'll be in the saddle more on a FG road bike than you would on a FW road bike. For you antique fans, my FG road bike is a Viscount (fork replaced). Feels more limber than my Vitus!
Good luck
Mel