Series Regulator or Switched mode?
Have you overlooked something here ...
The supply to the power supply can vary below the required output, the 7805/7812 type devices are 'Series Regulators', they are only capable of regulation by dropping volts across themselves, they are incapable of 'step up'.
The 5v rail is easy with a series regulator (you imply something around 3A for this), but for the 12v rail ...
Some form of regulator capable of outputting more than its supply is required if the input is to drop below the 12v out.
Consider a switched mode power supply, either producing the 12v directly or just stepping up the input (adding 5v or so?) followed by a 2Amp series regulator?. These devices are not particularly cheap or easy to build (but not impossible).