A 2-stage Allison engined Mustang would have had the same, or better range (The Allison, had a somewhat better Specific Fuel Consumption) and altitude performance as the 2-stage Merlin powered version.
At 27000 ft, with full ram, the best
ww2-vintage 2-stage V-1710 (with gear-driven auxiliary stage) seem to be good for 1200 HP; at same altitude, the V-1650-3 and -7 were making
between 1470 and 1490 HP. Almost 300 HP of deficit (plus a deficit of having lower exhaust thrust) does not seem to indicate that the altitude performance would've been same as what the Merlin Mustangs were making.
Another problem with the said V-1710 was that it was test flying in the last XP-40Qs in Spring of 1944 in the USA, while the 2-stage Packard Merlins were powering the P-51B,
in the UK, 6 months earlier, and were already trashing Luftwaffe by Spring of 1944.
Going with the 1st 2-stage V-1710s, of same generation that P-63A were using, means
another 200 HP deficit at high altitudes, and a worse benefit from the ram effect (see the note in the chart), and again lower exhaust thrust. Advantage of using engines of that generation has a benefit of being earlier in service, probably by the time of Big Week? Disadvantage is that a Bf 109G-6 with the fully-rated DB 605A is as fast at high altitudes, and climbs better. Fw 190 should be in the worse position, but the P-51s with the early 2-stage V-1710s would not be in the commanding position as it was the case with Merlin Mustangs.
There are already 2-stage Allisons in combat - the P-38s. With electronic Turboregulators, Dive Recovery Flaps, hydraulically boosted ailerons, and effective cockpit heating, (In real life, 3 out of 4), they were very effective air-to air fighters. In either case, you end up with what was achieved by the Merlin Mustangs - a fighter that could fly from England to Czechoslovakia, meet the Luftwaffe's fighters once there, and even if they could only flight them to a standstill, still be able to run them out of gas and then fly home.
Range was there, effectiveness was not. At least during the Big Week, P-38 force were claiming one-two LW fighters per day.
Merlin Mustangs were not fighting LW to the standstill, but were racking up the scores.
The basis was there in 1942 - The RAF's Mustang Is were flying Armed Recce missions over Germany, strafing anything interesting, and able to engage or disengage from the Fw 190s that they'd encounter.
Armed recon missions above Germany proper - how often were these undertaken by Mustang Is vs. the missions over the occupied Western Europe?