Whatever you get cheapest?
If you design the lathe using a belt drive, you can change the motor fairly easily at a later date.


All have pros/cons.
AC should be more reliable, however unless you go for 3 phase with a VFD, you won't get speed control.

Brushless in theory should be more reliable, as there are no brushes to wear, however I wouldn't say any of the cheap brushless options will be any more reliable. Have you searched out the homebrew spindle thread using a brushless motor for some ideas?