  We perform the energy minimization of the paired composite fermion (CF) wave
functions, proposed by M\"oller and Simon (MS) [PRB 77, 075319 (2008)] and
extended by Yutushui and Mross (YM) [PRB 102, 195153 (2020)], where the energy
is minimized by varying the CF pairing function, in the case of an approximate
model of the Coulomb interaction in the second Landau level for pairing
channels $\ell = -1, 3, 1$ which are expected to be in the Pfaffian,
anti-Pfaffian and particle-hole symmetric (PH) Pfaffian phases respectively. It
is found that the energy of the $\ell = -1$ MS wave function can be reduced
substantially below that of the Moore-Read wave function at small system sizes,
however, in the $\ell = 3$ case the energy cannot be reduced much below that of
the YM trial wavefunction. Nonetheless, both our optimized and unoptimized
wavefunctions with $\ell=-1,3$ extrapolate to roughly the same energy per
particle in the thermodynamic limit. For the $\ell = 1$ case, the optimization
makes no qualitative difference and these PH-Pfaffian wave functions are still
energetically unfavourable. The effective CF pairing is analyzed in the
resulting wave functions, where the effective pairing for the $\ell = -1, 3$
channels is found to be well approximated by a weak-pairing BCS ansatz and the
$\ell = 1$ wave functions show no sign of emergent CF pairing.
