I had a problem balancing Diamondbacks and original Ford 15" Tbird rims.   When I put the new tires on the shop told me they had a problem with two tires and they installed 5 weights on each tire.  I brought the two wheels/tires to another shop that fixes modern wheels.  They couldn't do much better.  They told me the tires AND rims were out a little.

There is a balancing machine called the Hunter Road Force.  This machine puts pressure on the tires while they are spinning and measuring the balance.  The machine can tell the tech how to "re-orient" the tires on the rims so the imperfections counter balance.  After they did this the balance was much better.  Went from 5 weights to 3.  Not many places have this machine.   I went to a Ford dealer and they charged me 1/2 hour labor each @$90.   My only other option was to locate "better" rims......   Old perfect rims are hard to find and the prices seem to be $75 and up......   

https://www.hunter.com/wheel-balancers/road-force-elite#overview

There is something to this, mounting radial tires on the original wheels that were designed to be used with tubeless bias-ply tires. However, a member of my Tbird club drove his 57 Tbird from Sacramento to Washington and back on radial tires on the original 57 wheels without issues. What can happen is that the "flex" can cause full wheel covers to come off. 

... and increased positive camber.

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Toe-in is slightly different on radials.

Were the tires mounted on 1956 steel wheels?

A radial requires a later style wheel. The earlier steel wheels may/will flex.

But if you felt bulging in the sidewalls, I would shake them, maybe try a warranty claim.

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Although I have not purchased either, Diamondback seems to have a better reputation. 

Not worth the expense for a cheap set of tires made in Mexico for a company in India. Tormel Classic. I will probably cut my losses and try Diamondback or Coker for a 1 inch radial with the bias ply look. What good is buying cheap off shore junk if you can't even use it safely? 

An out-of-round tire might not balance. It might be necessary to have tires trued, which involves shaving the tread, thereby reducing tread life. 

Would that explain why the shimmy seems to smooth out at higher speeds. I did mention that I felt divots in the sidewalls when the tires were new and I was scrubbing the whitewalls to get the blue soap off.

A steel cord belt that has not sealed properly in the manufacturing process will never balance out.  The steel will oscillate forever.  See if the beads/tread run perfectly straight on spin test.  You can see the oscillation/waver if that is the issue.  BTDT.  Joe-JDC  

JDC