Knowing nothing about your circuit, but having played for a few years with electrically operated Packard shutter controllers, I expect that you need the 1 ampere current draw for only a fraction of a second at a time. The solution may be a very large (electrically) capacitor, which will probably be considerably smaller (physically) than a pack of batteries capable of delivering the peak current that you need. To give an idea of the concept: I have a 12-volt solenoid which draws about 6 amperes. A 5 ampere-hour gel cell will operate it, but only for a fraction of a second before the voltage has dropped too far. Since a cylindrical solenoid exerts minimum force at the beginning of the stroke and maximum at the end, it needs a big surge to get going; two nine-volt batteries in series plus a 10,000 microfarad capacitor (gross overkill, but that's what I had lying around) slam the solenoid through its stroke with great vigor, to the extent that I should probably put some cushioning at the end. Given that the interval between operations is typically a minute or two, I could probably operate this thing with a stack of six lithium coin cells.
This approach may be recognizable to veterans of the flashbulb era, who used the so-called BC flash guns--- the name derived from "battery-capacitor"---to deliver from a small 22.5V dry cell battery the multi-ampere pulses needed to fire the bulbs. The big press-camera flash guns used three D-size dry cells to do what the BC components could do in a volume of about a cubic inch.
If you can go this route, you will probably find that your battery life is greatly extended; the large currents are flowing between the capacitor (which has a relatively low internal resistance) and the load, instead of through the higher internal battery resistance; your working lifetime may well be limited not by the remaining charge in the battery but by its ability to deliver enough peak current. Keeping the recharge current just high enough to accommodate the interval between operations would give the best battery life.
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