Going back to the original question there really wasn't much to chose between either the superchargers or the engines. They were just set up differently.
The Early Merlin was set up to give a critical altitude at 16,250ft. Using 87 octane fuel this ment that the engine could NOT use full throttle below that height without the risk of detonation problems. The lower the altitude the more the throttle had to closed until the engine was rated at 880 HP for take off.
There is no way to limit the amount of boost. The supercharger is geared to the engine and if the engine is turning a certain rpm, like 3,000 the super charger is turning a certain rpm. In the case of the Merlin III that would be 25,740rpm. It doesn't matter if it is at sea level or at 20,000 that is how fast the impellor will turn. Gearing a single speed supercharger to give good performance at 15-20,000ft means that it will deliver way too much boost at sea level and will require high power to drive it in the thick sea level air. Engines have to be built to withstand GROSS power. Net or shaft HP plus friction, supercharger losses and any auxilary requirements like pumps. It will also heat the intake charge reducing it's density while increasing the risk of detonation.
Using 100 octane fuel moved the detonation threshould allowing the throttle to be opened more at low altitudes.
The Allison's were set up with a lower critical altitude or full throttle height. THe early P-40 engine for instance was 13,200ft. THis ment that the supercharger was taking less power at low altitudes, and heating the intake charge less. THE C15 (-33) Allison made 1,090 HP at 13,200ft. 60HP more than the merlin at 3,050 lower. The differences would have been more if both engines were measured at the same altitude. The Melin would be loosing power at it went lower while the Allison would have lost power as it went higher. The Allison was rated at 1,040 for take off. the extra 160hp fover the Merlin III for take off and intial climb may have been useful in some cases.
At the start of the war most peaple's superchargers were capable of delivering a pressure ratio of about 2.3 at an effiency of between 60-70%. since you need a pressure ratio of 2 to 1 just to deliever sea level pressure at 20,000 ft that doesn't leave much extra boost for higher performance. Especially considering that many engines were already running a pressure ratio of 1.4 at sea level. 1.4 X 2 meaning the supercharger would need a pressure ratio of 2.8 to get sea level power at 20,000ft. After Hooker got through they were getting a bit higher pressure at 70-75% effecinecy. the higher the effecency the less power is needed for a given amount of boost leading to more NET HP. It also means less charge heating for given level of boost. Later they got pressure ratios of over 3 to 1 from single stage superchargers at good effiency. But with improved fuels the level of boost went up at sea level. 12lb of boost means a pressure ratio of 1.8 at sea level and a need for 3.6 pressure ratio to maintain power at 20,000ft.
A two speed supercharger gets 2 critical or full throttle heights but still runs into the pressure ratio problem. This is why 3 speed superchargers were never very popular. The extra speed would give more consistant power while climbing (an extra peak in the sawtooth chart) put it couldn't do anything about the pressure ratio limit. It aslo couldn't do anyting about the charge heating at altitude. Using large pressure ratios ment lots of charge heating.
Two stage superchargers sovled the pressure ratio problem and offered even more performance because a two stage supercharger will casuse less charge heating for a given pressure ratio than a single stage supercharger.