I don't think it matters very much, the details of the diode.
The difference between a Schottky (~0.3 - 0.4V) and a
plain rectifier (0.7V) in efficiency terms is modest.
Figure you will likely run unregulated and input-torque-
starved at 2A output:
Voutput = Vbat+Vrect
Eff1 = 12/12.3 = 97.6% (Schottky 0.3V)
Eff2 = 12/12.7 = 94.4% (PN 0.7V)
Either of these would seem acceptably good, to me.
I'd suggest a 5-10A, 50PIV rated diode with a tolerably
low high temp leakage (you could figure what is acceptable
from either the required charge retention time, or service
factor charge/discharge and an efficiency allocation).