Right way to disassemble radiator piping?
What I would do: 

Cut the long pipe. Move the radiator, making both cut pieces accessible. Remove and and discard both pieces of the long pipe. Unscrew the elbow-and-tee assembly from the radiator. Replace the short nipple with two short nipples connected by a union. Replace the long pipe with a longer pipe to account for the added length of the added nipple and union. 

This will leave you with plumbing projecting from the right side a little more than your current hookup, but it's better than pulling everything apart down to the first union on that line which may be, as you say, two floors below.

When you unscrew the elbow-and-tee assembly from the radiator, the short nipple will probably remain fast to either the elbow-and-tee assembly or the radiator. It doesn't matter which, just leave it, and add the new nipple to the other side of the union. 

You may need to separate the elbow from the tee in order to re-assemble the plumbing. If they are too rusted to come apart, consider that buying new ones is not really expensive and is a lot better than gouging your new floor. 

Edit: I just realized that you may be able to make the assembly a little shorter by screwing a union directly onto a street elbow. The order of the new parts (left-to-right) will be: new long pipe, tee with cleanout plug, vertical short nipple, new street elbow, new union, horizontal short nipple, radiator.