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The Rhythm King was offered around 1958 and 1959. This one is a 1958 model. All indications point to the Rhythm King as having the designation 1622B, which was a 1622T without trem.
There is one being demo'ed by a guitarshop on youtube. The controls are VOL+VOL+TONE (no treb). There was another one on youtube that had black fill plugs where the trem controls normally would be on a 1622T (where that was original Supro or not, who knows...).
See the Rhythm King schematic page.
Its looks like most of those Valcos use a 20/10/10 450V. (6161,6424)
Ken Watts at turretboards.com sells a S6424 kit, which is based on Supro's 6973 S6424 amp. Ken's schematic indicates 350V B+ on 6973 plates, and 23v on the cathodes with a 250ohm-R (250/23=.9, 45ma ea. tube?).
2 12AX7's: 300ma + 300ma, 2 6973's: 450ma + 450ma, (=1.05A)
The Classic-Tone 40-18062 ($68) is a 305-0-305 (firstname.lastname@example.org,5@2) replacement for valco PTs #3693, which were used in Valco 22, 510-24, 510-121, 510-149, 6161, 6400, S6422TR, S6622, and S6424.
If the OT is bad, the Classic-Tone 40-18063 is a 7.5KCT:8ohm OT for Valcos, however the Rhythm King original OT is a 7.4KCT:2ohm transformer.