[morzh]

The heat, if the regime of the radio is what it's supposed to be, is a normal way for a rectifier to run.
The thing is - you will have to dissipate that heat no matter what.
If you replace it with solid state than your voltage due to lower losses will go up, and yes, your rectifier may be cool but those extra volts will make the res of the tubes run hot.

So, you will need to engage a drop resistor to cut those volts. Guess what that resistor is going to do? Right you are, it will get hot.

So, chose your favorite place to dissipate. But you will have to do it.
Short of, of course, taking apart your transformer and reducing the secondary turns, and then engaging the solid state. That will do it. But do you really want to go through that kind of PITB?