As I recall, Rock Hill is 250 meter outdoor concrete track with 45 degree banking. Your TT bike would be okay for pursuit in the pole lane and probably accommodate a couple of gear changes. Otherwise the BB height is too low, potential pedal strikes on the banking even with 165 cranks, when riding on the banking a slower speed.
The other problem with using a TT bike on the track is gear changes. Horizontal dropouts on a track bike are generally a couple of inches to allow for changing gears with the same length chain. For example, I warm up with 80 gear inches and then change to race gears that may range from 92 to 102 gear inches. I can run all those gear inches on my track bike using the same chain without adding or removing links. And I may make 4 gear changes for a given workout.
And yes, racers use TT bikes on the track and in fact, Kevin M set a WR at Aguascalientes, I was there, on a TT bike frame set up for fixed. One has to accept the limitations of a road frame at the track. I do not know if he races that frame for mass start at the track.
FWIW, here is what I would recommend. Get a used cheap steel, aluminum or CF track frame but with horizontal dropouts that are at least 1.5 inches of usable adjustment. Buy a cheap set of track wheels and 165 track crankset. Get three cogs - 14/15/16 and three chainrings 48/49/50 all 1/8” do not get road width 3/16”. Put on a front brake for the road and remove it for the track. You should be able to ride 48/16 on flat road terrain and have 96 gear inches or less for pursuit at Rock Hill and you have a range of gear choices for the same chain length without removing links (pain in the ass) for the road and Rock Hill.
With respect to gears, you can start with one cog and one chainring and calculated how many gear inches you want for the road and then buy more as you need them.
As a final note, if you are planning to go to the track, do not go really cheap on the track stuff - think robust. At Rock Hill you will pull 2 Gs in the turns at 35 mph. Equipment failures (or racer failures) in the turns at high speed is a disaster. I will say no more.