1A Voltage Regulator - Maximum or output?
My awnser assumes that your battery can provide enough current / voltage.

You can use both voltage regulators, as the load only draws the current it needs. So if your load is drawing 1A constant, you can use any voltage regulator that has a output current of 1A or above. I would choose personally a voltage regulator a bit higher then the circuit draws. (So if the circuit draws 1A I'll choose a voltage regulator of 1.5A or higher) This is just to be sure your voltage regulator can provide the current that is needed.

Choosing between a low drop and a non low drop voltage regulator depends on the battery you are going to use. If your battery is like 15V a normal voltage regulator should do. Be sure to check the datasheet for the minimum input voltage when you are going to regulate it to 12V.