I'm no expert, but here are a few thoughts.

The track bike has a steep seat tube for a short wheelbase and a more forward hip position. Short wheelbase is for quick handling. Forward hip position is sometimes preferred for high cadence and sprinting. The road bike has a slacker seat tube for longer wheelbase and less forward hip position. Longer wheelbase is for stability and comfort, less forward hip position is sometimes preferred for lower cadence and climbing.

By using that setback post and sliding the saddle rearwards on it, you are sort of re-creating part of the road bike fit on the track bike. But even so it looks like the track bike has a shorter saddle to bar distance than the road bike. Shorter saddle to bar might be preferred for short, high effort sprints where you are pulling "up" hard on the bars to increase the downward force on the pedals, but too short and you might find your knees hitting your elbows or the bars on big out of saddle efforts. Longer saddle to bar distance might be preferred for spending many hours in the saddle, with room to flatten your back and stretch out comfortably.

If you put a really long stem on the track bike , you can get more saddle to bar distance. I'm not sure of the effect on handling, I think (not sure) it might slow the steering a bit.

You'll have to try it and see. Probably depends on if you will use the track bike as it was intended, shorter high intensity riding, or are planning to ride it like a road bike. My guess is the track bike is too small, but the proof is in the riding.