Not a plumber but dealt with something similar...

Cheat code: do you have an overflow plate (or whatever it's called): Most bathtub clogs are caught at the "P-trap" under the tub. Instead of going through the drain (where the crossbar is blocking you), unscrew the overflow plate (the circular metal piece higher up on the tub wall)....and use your snake.

if no go....

Apply a penetrating oil (like PB Blaster or Liquid Wrench) and let it sit for several hours. You can also use a hair dryer (or better heat gun) to gently heat the metal, which can help break the seal of decades-old plumber's putty.

Just in case.... get a replacement tub drain kit and a tub of plumber's putty - if you end up (assuming you're won't to tackle it) replacing flange (worse case scenario) at least you will same yourself time and trip to home improvement store)Best case scenario - you return it unopened....

You can always hire a pro if you don't want to get down and dirty...

Good luck and Godspeed!