I have a '56 Bird that has the flat-bar straps welded to the bottom of the chassis. I am the second owner of the car, have had it since about 1981..
When I took position of the car, I had known the original owner since she got the car as a birthday present in 1956. I took the car to a well known T-Bird Expert in Sante Springs, CA for a complete survey. I asked about the frame re-enforcement.. I was told that the car had an export frame. Reputedly Ford offered the export frame in 1955 on the Birds, discontinued the option in '56. Too use up the export frames in stock Ford randomly fed them into production on the '56 Birds.
I personally knew Alex Tremulus, a very famous auto designer that worked for Ford on the redesign of the Birds. Alex redesigned the '55-56 Bird to eliminate the rear end "lift" under speed. The continential kit added much needed weight to the rear of the car to improve stability at speeds in excess of 105 mph.
Alex redesigned the car for '57, changing the rear fender "fins" lengthened the trunk and reconfigured the fins to an air-spoiler/damn, which kept the air flow on the trunk and thereby created a downward flow increasing rear axle weight in lieu of lift like the '55-56 had.
If the steel bar was added to stop frame flexing due to the heavy rear mounted spare, the bar would have been added to the rear portion of the frame, in lieu of being on the front portion.