I like the idea of reusing or repurposing something old but in this case I am against putting a modern set in here. You will never find a flatscreen that will fit properly and you would have to build a mask to cover the gaps. The roundness of a CRT screen is what gives these TVs their character and you will never achieve that look with a modern screen in there. I suggest finding a big RCA console cabinet with doors that has all the insides missing. That way you can close off the modern part and have your vintage feel when the TV is not in use. Using that Westinghouse will just give you that homemade vintage frankenstein look that isn't flattering at all. I would suggest searching for examples of conversions before starting this project and going to the point of no return in destroying this amazing vintage TV.
Here is an example of a cabinet that was at the local repurpose store in Pittsburgh called Construction Junction. You could find one with a more modern furniture look to it.