So let's make this interesting and satisfy the OP. Take a Mitsubishi Kasei engine at 42L displacement, 780kg weight, and takeoff power of 1850 HP (with 87 octane fuel) and drop it in the Rare Bear airframe. The original engine for said frame was the Pratt and Whitney R-2800 Double Wasp with 46L displacement, 1076kg weight, and takeoff power of 2100 HP (with 100/130 octane fuel).
F8F stats per OTL have a loaded weight of 4627 kg with 2250 HP for takeoff. 454kg of that are in bombs, 275kg are in the combined *four* 20mm cannons with 325 rounds per inboard and 188 rounds per outboard cannon (total of 1026 rounds at 0.35kg per round, total weight of 359.1 kg [
http://www.pmulcahy.com/ammunition/autocannon_ammunition.html; look under 20x110mm ammunition]), and 61kg for each of the four 127mm rockets for a total of 244kg (
https://en.m.wikipedia.org/wiki/High_Velocity_Aircraft_Rocket). Range was 1105 miles with 185 gallon self-sealing main tank, not to mention an optional droppable 150 gallon fueslage tank or 100 gallon drop tank per wing (possible total 535 gallons, possible maximum range of 1835 statute miles [
https://www.thisdayinaviation.com/tag/grumman-xf8f-1-bearcat/], effective operational radius of approx. 450-600 statute miles).
So how does this change the aircraft? This proposed engine swap sees loss of 400 HP (17.78%) for a weight saving of 296kg. Removing the rockets saves 244kg while taking the bombs saves 454kg. Removing two 20mm cannons saves 137.5kg, decreasing the total 20mm rounds from 1026 to 600 saves a total of 149.1kg. This makes for a total of 1280.6kg off of its original loaded weight of 4627kg (27.67%); keeping the bombs <[or adding equivalent fuel of 166.5 gallons for additional range; 571 miles for just the fuel but say +500 miles to ferry range or +225ish miles to operational range to include the tank for total *operational* radius of 650 to 850 statute miles]> but still saving weight elsewhere as above still saves 826.6kg (17.86%). I am unsure how much more power the improved fuel would give the Kasei or what the lesser fuel would take from the R-2800.
In short you now have a heavy fighter able to tangle with any existant aircraft of the day with tactical air support capacity better than some frontline light bombers and possible operational range equal or better than the *ferry range* of the Bf 109 or Fw 190. As an example, at 400 mile operational radius from a base in Warsaw such an aircraft could reach Minsk, Riga, most of former East Germany, Vienna, or Budapest. At 600 miles it could reach Venice, Belgrade, all of Bulgaria and Denmark, western/central Ukraine, all but westernmost Germany, Stockholm, and Tallinn. At 800 miles it could almost reach Rome, Paris, or Constantinople. All of northern Italy, the Benelux, major areas of Scandinavia, Moscow, Leningrad, all but easternmost Ukraine, and most of the Adriatic would be in range.
