I have a Paasche H which is about the same thing as a Badger 350, and it draws paint just fine as low as 15 PSI with maybe 17 PSI being my typical painting pressure.
Since you're using a bottle, is the air hole in the cap clear? The hole has to be clear, to prevent the air pressure in the bottle from dropping as paint starts to get pulled up the tube and thus preventing the paint from getting up into the airbrush.
Have you opened the tip enough? Unscrew the tip anti-clockwise to open it up an allow more paint.
As another experiment, if you do think your paint is thin enough, try shooting straight water through it. If it can spray water, but not the thinned paint, that would tell you you need more thinner. If water doesn't go though, then something else is at fault.
Andy