I do not believe that you will have much success trying to defeat the MiniMe's behavior when battery voltage drops to the unit's minimum threshold. I also question the wisdom of repeated running the unit to the drop out point - and i think you agree and that is what your post is about. I do believe the behavior to be within the normal range and agree with rustoleum and todd e that you can best avoid this by running a battery with more capacity and know your equipment.
Now, if you want to protect the unit, i would build a circuit that shuts off power based on a voltage setting, not based on current demand. it would be more precise that way, or you could just use a small battery voltmeter to track the battery voltage and shut it down yourself.
The minime runs on 6-16 v da at 5.5w, so at 12v dc is pulls just under 500mA. your 1.2Ah battery is going to run for just under two hours before it has reached its max safe discharge point. i would go to a larger capacity battery, unless the 1.2 was a mistype and you meant 12Ah. The ideal battery would be a 10Ah 12v battery as the 20hr rating indicates the batt would run for 10 hrs at a 500mA draw. so in fact one might get a little more than 10 hrs. however, this is a heavy device to lug around... so looking at choices, maybe go with an RC pack @9.6. (good for the budget) or go further and run LiION like i do (costs more but very compact and very light).
TECHNICAL NOTE: when an SLA battery is powering something at more than its 20hr rating, battery capacity is diminished.
EXAMPLE: a 12v 6.5Ah battery's 20hr rating is 325mA draw for 10 hrs and thus has a rated capacity of 6.5Ah. BUT when you up the draw to 650mA (2x 325mA), the capacity falls to 5.85 Ah and the battery only runs for 9hrs. SO, we have doubled the draw and unfortunately will receive less than half the run time, 20 hrs vs 9 hrs.
THIS is important to note because it indicates that you cannot just do straight math to determine the runtime of an SLA battery. and the more you currenrt you draw, the more you shorten runtime and reduce battery capacity.
neil in san marcos