The rectifiers must be rated *twice* the no-load DC.
Since they used 350V+350V worth of caps, we should assume there could be as much as 700V DC, and must use 1,400V diodes.
The cheapest "1,400V" diode is two 1N4007 in series. We see this on the preamp supply, so it makes sense.
If you must back-up the bottled rectifier with sand-state, this also must be 1,400V to hold things "if" the rectifier goes short or gassy.
Yep. The measured B+ at 600V means that a 1200V diode is marginal. I'd like to see 2kV in an equivalent situation in my own amps if that happened.
Mouser has the 747-DSA1-18D which is 1800V, 3.6A, but it costs $4.56 each. There are probably other choices in the industry. I didn't do a massive search. You're right, 2x1N4007 is cheapest, but this setup needs help.
It is important, but often ignored, to provide some equalizing of the reverse voltage on series devices for high voltages. That's what the resistors in parallel with series diodes do. It's a good first order fix, and forces the reverse voltage to be equal across, for instance, two series diodes, by providing a leakage path that completely swamps out the differences in reverse leakage of the two series diodes.
That works fine for DC, but what about transients? In that case, the reverse voltage divides by the inverse of the ratio of the diode capacitances (and inductances, which I'm choosing to ignore for the moment). In that case, you need to put a cap across the diodes that is greater than 10x the reverse capacitance of the diodes to equalize them for AC conditions. If you use a low-R + C snubber, often the snubber can do this for you. But it's something most people haven't run into.
There's a lot of discussion online about various ways to "bulletproof" tube amps--fuses on the B+ HT voltage, fuses on the heaters, etc., backup rectifiers, circuits to automatically sense/adjust bias voltage, etc.
Yeah. A lot of it from me.

I like the idea of solid state minions making the world safe for tubes.
I doubt adding diodes would have much effect on the efficiency of the 5u4, since they'd be wired before the tube.
There is substantially zero effect. The tube rectifier has a 30-60V forward drop, the solid state a 0.7 - 1.4V drop, and the same current, so the power loss is in the ratio of the voltages. Call it about 2% more loss in rectifiers only for a SS diode in series with a tube.
There are faster diodes, of course, but I'm sure the 1N4007 has a shorter recovery time than the 5U4. The SS diodes might introduce some noise spikes, etc.--maybe the tube's response would smooth those out...
The 1N4007 would follow the tube down in conduction, so the spikes and ringing from a standard silicon diode slamming off would not happen when it's in series with the tube. The turn-off transient only happens when normal diodes quit conducting abruptly and excite an RF ring in the associated wiring. Snubbers can largely eliminate this; this is why you often see caps and R-C networks around rectifiers. Fast+soft turn off diodes also prevent it without snubbers. So would a series tube.

Or maybe I should take this a sign that those 200uF 350V filter caps might be failing after all? But the amp sounds great right now, none of the classic failing filter cap symptoms. Argh! I just don't know for sure...
Simple enough. Open up the wires after the filter caps to the rest of the amp. Clip meter leads to the caps, get your hands out of it, then turn it on, and off. The B+ will go up to the no-load voltage, then start decaying. It will decay at the R-C rate of the caps and equalizing resistors.