Driving power MOSFET from 3.3v or lower Arduino Pro Mini
I don't think the mosfets will be an issue. Looking at the last one (BSS806NE); in figure 5 in the datasheet you see that for 1A and 1.4v (Vgs) you'd get a Vds of only .1v. 

I think a bigger problem might be the voltage of the batteries dropping under load. You'd have to measure that with a multimeter while the motor is running. A simple cap might help with a little noise, but not with the voltage dropping while the motor is running. Not sure how much the AVR would mind having its supply voltage dropping all of a sudden.

Of the top of my head the minimal voltage of the Atmega328 @8mhz is 1.8v, while default fuse settings use 2.7v, so I'd probably change to fuses to make BOD at 1.8v. Other fuses don't need to change.

Backup option sound good, though you might have to go with 4 x 1.5v depending on the voltage drop under load. Another alternative might be to have two sets of 1.5v batteries in parallel, halving the current load on each battery.