First you need to do some reading on how to properly mount a bath tub.
Bath tubs are supported from both their bottoms and the the upper rim.
A 2x4 frame is used to support the upper rim in most new construction homes where I live.
No, that little wood will not be up to the task of supporting a bathtub full of water and a person. You would be miles ahead to use metal in this application to keep your lost height to a minimum.
"Don't argue with stupid people. They will just drag you down to their level, and beat you up with experience."
Patently waiting for the apocalypses to level the playing field in this physiological game of life commonly known as Civilization