1st post so don't flame too badly.
The requirements from specification - X-608:
twin engines - X-609 is the equivalent specification for single engine
1k usable payload - flushes out to 2 - 0.3" M1919, 0.5" M2 (200rpg) and a 23mm Madsen cannon (50 rounds)/25mm Hotchkiss (Just cannon was specified in requirement, but the those were leading contender to be designed around).
Tricycle gear (for bonus points)
360 mph @ 20k'
fly at full throttle for an hour.
take off & landing with 2,200' over 50' obstacle.
Allison's new V-12
Turbo super chargers - they were going to be need anyways.
Allison V-12 with mechanically supercharged made 960hp on 100 octane fuel at 12k' in '37 when the Lockheed team started the design. RR Merlin basically made same 950hp at 11k' on 87 octane fuel, but is down to less than 800hp by 20k'; Allison loses power faster as it is less efficient. (Just 2 yrs before Merlin was only making 740 hp at 12k', but Rubbra and H**ker increased the supercharger efficiency from 36% to mid 70s. I can't but wonder if Allison, General Electric, Pratt & Whitney, Wright, etc didn't benefit from RR technology to improve their superchargers as they seem to also make some dramatic improvement without any formal recognition). With turbosupercharger, there was ability to cool the compressed air between the turbo and mechanical supercharger, so setup in XP-38 makes 1,100 hp (engine would actually make more but the radiators couldn't handle any more). So, your design is going to have to fit engine and turbo in close proximity. You also need to find some location for main landing gear - again ideally in close proxity. Engine is largest single mass, so if landing gear is any distance away, you will have to add structure - read weight. So, you aren't going to get DH Hornet short nacelles if engines are in wings. Also, in '37-39, the Meredith effect isn't know, so no leading edge/Mustang radiators, heck not even the YP-38 podded ones. Burying the radiators in wing ala Westland Whirlwind leaves no space for the >400 gallons of fuel in wing, if you're putting it in central fuselage, you're making it big = heavy/lots of skin drag. So, a "conventional" twin doesn't happen. Strike figure 1.
The armament requirement, specifically a cannon like Madsen. At ~115lbs for gun only - which means about 175lbs with all accessories (muzzle brake, trigger, etc but not ammo). About 10% heavier than HS.404 for reference. So, we definitely don't want to pack 2. While a belt feed was available for the Madsen, the 50 round drum is the most common configuration. So, if we attempt a P-82 layout, we will need to install it in center of the inter fuselage wing, and we will have an UGLY, not aerodynamic lump for the breach/feed mechanism and probably another for the drum magazine e.g. Spitfire with early HS.404. And we still need to figure out how to make the P-82 configuration tricycle gear. Strike figure 5.
The Allison engine isn't designed for cannon to fire through hub ala DB 601/605/603, Jumo 210/211/213 or Hispano Suiza 12Y engines, so a Dornier Pfiel/Fokker D.XXIII/Bolkhovitinov Sparka/Kawasaki Ki-64 configuration isn't happening. But something like the Bugatti Model 100 would certainly check the boxes. P-75 had cannon thru the remote gear box along with 4 - hmg, just install engines in tandem rather than the W-3420 of the P-75. Raising the cockpit to fit tricycle gear helps with rear visibility, not a bad trade off if you need the bonus point. Probably a very nicely looking aircraft possible - and they say if it looks nice, it flies nice. Probably better than historic - fewer joints for interference drag.
Options 2 and 3 are possible, and Allison certainly had no concerns with remote propellers - P-39 Aircobra, P-75 Eagle, B-42 Mixmaster. There are some issues with packaging which add weight but nothing insurmountable. I consider pilot in nose a preliminary design, putting between engines ala Ki-64 or behind like Sparka are possible.
The wing for P-38 is a plagiarism of that just designed for the Constellation Excaliber - same basic root and tip profiles, just a smaller planform and root a little thinner. Saved a ton of design calculations. Size of wing is based on loads of all components, which don't really change in any of the scenarios, so wing won't change significantly.
Bf.110 started flying with 600hp Jumo 210s - needing more wing area to fly at same weight isn't surprising. Messerschmidt's Zerstorer needed as re-wing like the 109 received between E & F models, between the B (Jumo 210) and C (DB601) versions - they were strengthening and aerodynamically cleaning it up anyways.