I appreciate and applaud your plan to ride on your local velodrome non-competitively. I used to regularly attend "Intro" nights at mine until they told me that it was time to stop it and move into their regular racing events. I didn't

The Prercursa that seau grateau mentioned is a well-regarded bike around here, that in the past was a pretty decent price, and nowadays they've added more customization "options" than they once offered. But you have to to factor in that shipping costs from GB have gone up a lot lately and also that you may have to jump through Customs hoops to get it in here, that may even likely even add import fees. Felt TK3 is also a nice entry-ish level real track bike that if you are the right left-over size you can find at big discounts on the interwebz.

If last years Fuji Track Pro is your heart's desire - there's still a couple on-line retailers still selling marked-down '18s. There are others, but IIRC City Grounds has a pretty good rep...
https://www.citygrounds.com/products/fuji-track-pro-usa-bike-2018?variant=95331745817&gclid=EAIaIQobChMImpjpou674AIViZyzCh3TngKmEAkYBCABEgKAJPD_BwE

Have you thought about inquiring in the real Track Racing sub-forum here? Check in the Racer's forum.

Have fun & good luck...


Edit...Dammit, I'm too long-winded lately - PeopleAreIdiots and Philasteve beat me to it...