  We investigate the electronic structure of alternating-twist triple
Bernal-stacked bilayer graphene (t3BG) as a function of interlayer coupling
$\omega$, twist angle $\theta$, interlayer potential difference $\Delta$, and
top-bottom bilayers sliding vector $\boldsymbol{\tau}$ for three possible
configurations AB/AB/AB, AB/BA/AB, and AB/AB/BA. The parabolic low-energy band
dispersions in a Bernal-stacked bilayer and gap-opening through a finite
interlayer potential difference $\Delta$ allows the flattening of bands in t3BG
down to $\sim 20$~meV for twist angles $\theta \lesssim 2^{\circ}$ regardless
of the stacking types. The easier isolation of the flat bands and associated
reduction of Coulomb screening thanks to the intrinsic gaps of bilayer graphene
for finite $\Delta$ facilitate the formation of correlation-driven gaps when it
is compared to the metallic phases of twisted trilayer graphene under electric
fields. We obtain the stacking dependent Coulomb energy versus bandwidth $U/W
\gtrsim 1$ ratios in the $\theta$ and $\Delta$ parameter space. We also present
the expected $K$-valley Chern numbers for the lowest-energy nearly flat bands.
