The T500W is not Airprint ready so that’s why it’s not showing in the app. (Apple has a list here.)
iOS devices require Airprint ready printers for it to print from any app. The Brother 510w is the one that has Airprint.
If it’s newly bought, it‘s possible to return it.
In many cases, the merchandisers in the stores do not know the difference between Airprint and wifi printing, so be sure that the box says Airprint or shows the Airprint logo.
We have a list of Brother printers here.