There are quite a few powersupplies that fit the bill, you probably want to start by looking at the PicoPSU series. I believe you should be able to solder a ATX pigtail to the row of pins just behind the square din power jack, on my board theres two layouts for the filter coil, and the larger layout is unused leaving a nice row of pins with +/- 12v, + 5v and GND. (no idea on what order they are in, check the A1200 schematics on aminet)
There is atleast one of the PSUs in the series I mentioned, or a similar series available on mini-itx's store available in a Wide-Range input, capable of operating between 8v and 24v input. Once the battery discharges to 8v, it will cut off to prevent deep-discharge of the battery. Assuming you have a non-expanded A1200, you should be able to fit a reasonable sized battery, picopsu, and related wiring inside the stock case.