Installing pull-up bars. Will this design hold?
4-each on each end of 1-1/2"x #12 screw is plenty to hold the body weight of a hefty adult male. No need for second layer of lumber.  No need to extend through the 2x6 into the stud. Your 2x6 needs to be secured to every stud, 16" on center, with at least 3" construction screws; not sheet rock screws. 

The screws you show from Lowe's would be a good choice for the flanges.  They don't need to be that long.  Predrill your holes to the size of the screw shaft, not the thread diameter.  With predrilling the screws will hold with more strength.