What's better about a 5AR4/GZ34?
Lots less voltage dropped across the tube, lots. 5V4 is another good tube to use for low-rectifier drop, and sometimes cheaper than GZ34s.
http://www.300guitars.com/articles/rectifier-tube-voltage-drop-chart/ Ironically, a 5U4GB (if you believe the linked chart) drops 15% more volts than a 5U4 *GA* (Who'd a thunk it?)
I'm not sure I believe that. You seem to be losing a mere 23 volts across your 5U4 [GB, I assume]
I am also not understanding why the "mild" changes you have going on are reducing the 6L6 current so much. But, I suspect it is because you are switching on and off the filament of your 5U4. More specifically, because you are switching only one side of the 5U4 filament and leaving the connection across the (cold) 5U4 filament from the (now SS powered) B+, thru the filament, and thru the 5v winding. Yeah, maybe it sounds goofy.
I would try these:
1: Switch to solid state mode and pull the 5U4 out of its socket. See if you get any noticeable changes in voltages. Easy.
2: Temporarily, break "the connection that never goes away", the connection where the filament of the 5U4 goes to the top of the filter caps by jury-rigging an SPST switch in there. Measure. <<--This is just for temp use. Use a heavy-duty switch like you might use for switching line volts, not a cheesy little 1/4" bushing pickup-selector-sized thing. Just hang the switch up in the air and put some tape around it or safety it in some way.
3: Goofiest suggestion of all: Reverse the phase of the 5v connection to the 5U4.
OK, now you can let the laughing die down.