1" BSF thread is 10 tpi with a depth of 0.064" so the core diameter is 0.872" which differs from 0.750 (3/4 UNF) by 0.122, leaving a minimum wall of only 0.061"
Which is not much metal to transmit the torque when you are cutting hard.
I see little alternative but to follow Duncan's advice:
Bore out the chuck and press / Loctite/ Screw and Loctite in a larger bush, to provide more material outside the 3/4 UNF thread, when you cut it and the register.
The bush would need to be 1 " diameter, if plain, or 1.250" if threaded, to make me happy.
But I favour brick mausolems!
Howard