On the basis that you use a switching regulator, the 12v 4AH battery will provide a longer duration than the 6v 4AH battery.
This is because the switching regulator "transfers" watts rather than a linear regulator which simply "transfers" volts. Note that the word "transfers" is a simple metaphor to illustrate the point.
The 6v battery stores 24WH of energy and the 12v battery stores 48WH (volts x AH)
Battery capacity is usually related to a discharge period of 10 hours, so your 4AH battery discharged at C/10 should deliver 0.4 amps for a period of 10 hours. If you reduce the discharge rate to, say C/20 you could expect to get around an extra 20% of capacity. This 20% gain factor tends to repeat for each doubling of the C factor.
Your quoted discharge of 200mW out of an 80% efficient regulator implies an input wattage of 200 / 0.8 = 250mW. For a 12 volt battery, this is a load current of around 0.02 amps. Compared to the rated battery C/10 current of 0.4 amps, this implies the battery load is C/(10 x .4 / .02) = C/200 which is very conservative. At such a low C loading you might expect the battery capacity to rise substantially, based on 4AH x 1.2 (C/10 to C20) x 1.2 (C/20 to C/40) x 1.2 (C40 to C80) x 1.2 (C80 to C160). That all works out at around 8AH.
That's the good news, now the bad :
If you are using lead-acid type batteries, these shouldn't be discharged lower than 50% capacity and ideally no more than 20% of capacity rating. Depth of discharge defines battery life. Deep discharging reduces battery life drastically. So, assuming you limit consumption to no more than 20% of capacity (1.6AH based on the increase to 8AH above) and assuming a load current of .02 amps, your battery duration will be 1.6 / 0.02 = 80 hours.
Edit : Last paragraph edited to clarify discharge limit