At least in the mid-1930's Ford did put some identification markings on at least some of the radiators produced. I've inspected many radiators for 1935-1936 trucks and while most have no markings, some do. I have a Ford engineering drawing which specifies the marking requirements and also have at least two examples of radiators with identifiable markings. I don't know how many radiators got ID markings for all of 1932-53, but some did. Here are two examples of a 1935 big truck radiator. One is embossed "51 D" (Model 51, Dearborn) on the upper right-hand back side of the top tank and the other has a soldered tag with "51A L" (Model 51 (PN 51-8005-A) by Long Company). The codes for the bottom letter under the line indicate where the radiator was produced: D - Dearborn (Ford), G - Green Island (NY) (Ford), L - Long Company, Mo - Modine, Mc - McCord.
This information might not help your exact situation, but I figured it might help somehow or at least give you something to go on for the future.