Ah okay I get it now, wrobel's comments are correct. The exact formula in the approximation of b/R -> 0 gives the given answer. I wasn't being consistent with my coordinate system and neglected yet another contribution to the angular momentum (due to motion of center of mass), the horizontal and...
Sorry I'll clarify:
"Because the object precesses, it has a vertical contribution to its angular velocity [oops]. For that contribution to the angular velocity, the angular momentum due to it isn't vertical but rather rotates (thus needing the torque to do so), because it doesn't align with the...
Because the object precesses, it has a vertical contribution to its angular momentum, whom their contributions to the angular momentum isn't vertical but rather rotates (thus needing the torque to do so), because it doesn't align with the principal axes. Thus why can I assume these are...