I bought a aftermarket aluminum radiator for my 250 powered '37. It fit reasonably well but not a direct bolt in replacement. My stock radiator cooled OK, even in 100 degree weather it never go hot enough to really worry about. I changed due to the top tank always leaking. The brass was so thin, about once a year the seam would crack due to heating cycles. I finally got tired of pulling it out and repairing it. The new one cools really well, it never gets over thermostat temp, ever!
If yours is in good shape, it will cool just fine as long as you are not running a super high horsepower engine. Stock engines are just fine with old style radiators. You should have it converted to a pressure system and run no more then 4 psi cap. The old cores and brass ends don't like to much more then that. With a open system like stock '37, the coolant that gets pushed out when hot, never gets drawn back in, and over time you can start to run low.
With either a 350 or 250, the radiator and or fan will need some work. Mine has the crank in the same place as the 216 was so the fan blades hung lower then the bottom tank. I have lowered the radiator as much as possible and moved it forward as much as possible and still only have about 1/2" clearance and the blades still go past the core about 1" . Be sure to have the radiator installed and mounted when placing the engine and trans in the frame. I suspect the 350 water pump will also set lower then the 216. If you set the engine high and back, the radiator problems are a lot better, but you end up with a hugh transmission tunnel in the cab, and you really don't want that. The throttle pedal starts to close in on the steering column and brake pedal, ask me how I know that!
E-mail me if you want some pictures of my cooling system or engine locations. Maybe I can help you where I messed up!
Joe