Indeed, the diodes were not good. It is quite possible that 1N5399 would be just fine for the task, but the specimens installed are sub-quality.
I had to enlarge the holes, and have installed BY255 instead: 1300 PIV and 3A continuous current. No problems with the diodes since, with or without cap.
I had to externalize the fuse - but I did blow several fuses. In the end I understood that the issue was with the way I started it - without switch, by inserting the plug into the socket. I expect that with a switch this would not happen... but it seems that a fuse is always a good idea.
Modification: the original unit had 67 primary and 7 secondary windings. I chose to avoid messing with the primary and concentrated on the secondary.
First tests have shown the usual predictable results - very bright light from the 55W bulb, and even brighter light with the cap. The increase in DC voltage was basically from 208 to 312V (330uF cap), and the increase in measured HF AC (with my DMM) was 1.8x. Original 24.5-25V, with cap 45V approximately.
Assuming 11.5V or maybe slightly more, I have decided to go for 2x4 turns, expecting something between 9.6 and 10V per secondary. The wire I intended to use was too thick (insulation, probably), thus I used some (Litz) wire 1mm^2, AWG 17 approximately. AWG16 is recommended for 0-10ft in critical applications, thus I expect that at maybe 1/2ft AWG17 Litz would be fine for HF AC). I wound one brown and one blue wire parallel to each other (minimizing stray capacitance between the windings, and improving efficiency) into two separate windings which are basically spread across almost 300 degrees - at first with 5 turns.
The result, each with 55W bulb, without cap, was rather dimmed light (11.5/7*5=8.22V) but with cap the light was again very bright (11.5x1.48/7*5=12.15V (It measured 26.5V on my DMM).
Since it is rather easy to remove one turn (just extract one turn, I do not even need to unsolder anything), it came back to my original intention, 4 turns per secondary. With the cap, lights were dimmed, but somewhere between standard light and 5 turns without cap. DMM measurement almost 23V, and basically in accordance with my expectation: 11.5x1.48/7*4=9.72. Measured with the toy meter, interestingly 4.9, just slightly lower than 5 (which is what I measure with the units I am using in the RH813). With 5 turns, at estimated 12V, the toy meter was at 6V. It seems that if current draw is approx. 5A, it is going to show approximately 1/2 of the expected RMS reading.
Finally, the 55W bulb is probably drawing a little more current than the heater of the 813. All measurements and testing was performed with my standard 223V home mains, while once I try them in the RH813, they will operate at 230V (boost transformer). Thus I expect that the voltage per secondary will be slightly lower than the units I am using, and probably ranging between 9.7 and 10V. Being in the 5% tolerance (on the lower side), I guess that is totally acceptable.
What remains to be done is to test it on the amp. That will of course have to wait a little bit. I am interested to find out: 1) voltage values compared to currently used units; 2) sound: hum/noise/buzz issues (if any), and of course how does it sound, i.e. whether there are any differences.
Although this unit is declared as operating at "just" 30kHz, I do not expect to be able to hear any hum/noise/buzz, since I am not hearing any with the units I am currently using. As for the sound, I guess one HF AC unit is equal to another.. or is it? This is going to be a test of a two secondary unit, but I cannot imagine that using separate transformers for each filament should lead to particularly audible differences...
Anyway, if everything goes well, one transformer and one cap cost at least 2x less than two transformers... and you can always keep the second unit as a spare
Not to mention the space, since these units are rather small anyway.