LM338 will give you a 1.2-12V at up to 5A variable supply. You'll need a huge heatsink for 1.2V @ 5A unless you set the input voltage to something sane.
You can use a 4.3V zener diode to cause the LM338 ADJ pin to track the input at 4.3V below the input.
Since the LM338 output stays at 1.25V above the ADJ input, the output will go from 1.2V to over 12V when the input is adjusted from 4V to 16V.
The regulator power dissipation will thus be basically constant with change in voltage for a given output current, whereas otherwise it would go up as the output voltage goes down.
The LTspice simulation of such a circuit is shown below for an input of 4V to 15V.
The output goes from about 1.3V to 12.9V.
R_Bias provides the minimum 50mA output current for the regulator to maintain regulation.
(The circuit shows an LM317 but an LM338 should work the same).
Boy, these are all terrific options, so being the geek that I am, I am gonna try them all.
I think that I have been infected with a "if you can build it, do it", virus.