Is this a reasonable circuit for locked antiphase PWM?
Rushing ...

Antiphase PWM per motor looks OK.

Disabling IC using EN lines will often create large motor spike - as long as internal IC catch diode rating is OK that's OK. Otherwise add external diodes to supplies.



What battery chemistry. What size?

AA Alkaline will often tolerate significant regeneration.

Unless battery is small physically or you want exceptional shelf life then NimH will be better than Alkaline. For very long shelf life LSD NimH are getting very good.

At AA size, energy density of NimH now equals or exceeds Alkaline and high load dishcharge characteristics are much superior so there is little reason for Alkaline.
 At small cell size Alkaline MAY make sense. 



I have not tried this
 Answer is "seems like this to me".
 Caveat emptor :-):

Motor inductance and resistance will set t = L/R time constant. But, very easy to put small resistor from motor to ground, apply voltage and watch voltage in resistor and thus current rise on scope. PWM period which is small compared to total motor rise time will (probably) be good.