The Army may have been slightly obsessed with the Turbo-charger but not to the extent you seem to think.
They did test (pay for) at least one two-stage supercharged plane in 1939.
Work on the turbo for the Allison started in early 1933. GE (not Allison) got the $10,006 contrat to supply a turbo in 5 months. this early model was only supposed to provide sea-level air pressure to the engine supercharger up to 20,000ft.
By 1937-38 progress was not good enough for service work.
Allison did start work on a mechanical 2 stage supercharger on Nov of 1938. They did get experimental versions under test by 1940.
Now in 1940 they were still trying to improve on GE's rather poor performing superchargers, which Allison did. Just not the extent of improvement that R-R was getting per impellor.
as an example Allison was getting 1150HP at 3000rpm at 21,000ft using 49.7in of boost. This doesn't sound bad except for the fact that the engine was developing around 1630 Indicated HP. 147hp for friction, oil pumps, coolant pumps and other drives, aproxomatly 120 hp needed by the engine supercharger and about 220 HP needed to drive the auxilary stage. This early test engine also used a single speed drive for the aux supercharger. This was soon changed to a variable speed hydraulic drive. The first set up was intended for the Curitiss P-60 while the second set up was actually flown in Bell XP-39Es.
"The fuselage was lengthened by 1.75 feet to accommodate the longer -47 engine."
"Empty and loaded weights were 6936 lbs and 8918 lbs respectively, making the XP-39E the heaviest of all Airacobra variants. During tests, a maximum speed of 386 mph at 21,680 feet was attained, which was much better high-altitude performance than other Airacobra variants. An altitude of 20,000 feet could be reached in 9.3 minutes."
"About two weeks after its maiden flight, the first XP-39E crashed during spin tests on March 26, 1942."
These quotes are from:
Bell XP-39E Airacobra
It took R-R 15 months to do the work on the two stage Merlin and they already had the Vulture supercharger sized for about a 1700HP air flow at altitude.
So in 1938 the Army would have had to figure out that the Turbo wouldn't be ready even 4 years in the future, and try to speed up Allisons work by well over one year to get any numbers into service by spring of 1942.
Allison would not only have to figure out the configuration but also substationaly improve basic compressor perfomance. Improving compressor efficency has the triple benifit of requiring less power to drive the supercharger for a given airflow (power) meaning more power to the prop. Lower heating of the charge temperature meaning higher air density and more power. And the lower charge temperature also offers either a higher boost limit or a greater safety margin from detonation. In a two stage system a more efficent compressor also means you can use a slightly smaller intercooler.