Good question - maybe help clear up some of MY misconceptions.
I've always gone with the theory that both modes present the same load to the amp, and any volume change is due to the triode mode's relative inefficiency compared to pentode. So ... why triode? Because some find it more sonically appealing than pentode for certain genres. Jazz seems to be the hot ticket item for that. I don't much use it myself as I prefer the punch of the pentode. Maybe it's a volume thing - I tend to push it to what the aforementioned "some" might find extreme.
That said, the GZ34 has always been a bit underpowered for these amps, working right at the limit of their design. If you do decide to go with bottles, I highly recommend the GZ33 or GZ37. I got a deal on a NOS GZ37 a while back and it's my #1 choice now, driving a quad of KT120's. Amp just sounds a lot more fluid now and doesn't seem to be straining at all like it did with the GZ34. It also sounds somewhat less clinical than the copper cap.
Probably worth mentioning - I got an ST120 ... same thing, only different ...
*Also, I don't believe you should be seeing any drop in the bias when switching modes, at least nothing that should be a concern. You should be able to just flip the switch on the fly under load without having to make any other changes. Maybe an out of spec resistor in the PT network?