Got the box knocked together. I simplified the design by not having a switch to go from the higher voltage to the lower, instead I'm just using an outlet with the two plugs going to the different rectifiers. The two transformers work exactly as I expected and the voltage drop across the rectifier brings the second output to exactly 5V ±0.2, the upper is 24.5V ±0.2. These transformers are good for 200VA, so 40A at the lower voltage and 8A at the higher.
I made the plug from an old battery charger, ironically, and spliced the plug and clips together, which makes it about 12' long. The entire box is a good 20lbs.
The next step is to find something rusty and put it in a bucket of solution.