The Plymouth business coupe used the same body shell from 37 through 39, so the front sheet metal should bolt on, when used as a complete unit, there is probably a difference on how the fenders attach to the running boards. The individual parts like fenders, grill, hood, and hood sides are all different between each year, 37, 38, and 39. The 39 did use a more modern chassis with the independent front suspension, but you will get rid of that when you transplant the Dakota chassis.

I suggest you use the Dakota front clip rather then the entire frame, The Dakota frame kicks up pretty high in front of the gas tank and never comes back down. If you use the Dakota rear frame, you will end up raising the entire floor from behind the seat to the rear of the body about 6". But if you need floors anyway...with care, the steel box floor from the Dakota makes a pretty good rear floor pan for the coupe body. I did that with my 48 Plymouth coupe, and I have a few pictures.
The 39 frame and the Dakota frame join nicely at about the firewall.

As far as the 92 Dakota drive train, I would bet the trans is the 500 base OD, and the rear axle is an 8 3/8 (formally called an 8 1/4), it likely has 3:55 gears, and there is a good chance it has limited slip, but it also has 6 bolt wheels. The front hubs are also 6 bolt, but those can be converted to 5 bolt by buying new 5 bolt rotors. Everything else still works. As far as the trans, it won't care what you use to shift it, if you have an old slap stick, throw it in there.

AC, The Dakota heater/ac box is too wide for the coupe, however, the right end has about 6" of recirculating air duct work that can be modified. I was able to use the modified Dakota heat/ac unit in my 48, but I think the 48 is wider then the 39, you will have to measure. Using the Dakota heat box, I could connect the AC if I wanted to. There are some really nice compact heat/ac street rod units out there, but they ain't cheap, check Vintage Air. Most of the aftermarket heat/ac units are well designed, sort of a follow the directions and plug and play, there is little engineering required. Using the Dakota heat/ac will require some engineering.

My 48 Plymouth business coupe was a quick build. I parked the Dakota side by side with the 48 Plymouth, front wheels to front wheels. I made note of what had to move to line everything up from the front wheels. I noted where the firewalls were, where the motor had to set, how much the Dakota frame needed to be shortened, and where the rear end had to be. Then I removed the Dakota sheetmetal, and relocated what ever had to move to the proper location. I used nearly everything off of the Dakota, including the passenger floor pan, the firewall, the wiring & computer, the dash, the heater box, the steering column, and the box floor. I used most of the extra sheet metal as patch panel material.

Once everything was in its proper location on the Dakota modified chassis, I braced the coupe body, and cut the remaining floor out of it. I lifted the body, and trimmed the width of the Dakota until I could set the coupe body on the chassis. Then I went about connecting the Dakota floor pans to the coupe body shell. Remember all that extra Dakota sheet metal? The whole project took about a year to make it drive able (but not pretty), between working on other peoples stuff (to make a living).
At this point, this is the car's 6th summer on the road and we are just past 48,000 miles. Its a great driver. Gene