Maybe instead of mixing resistors, you have a SPDT switch to connect the output of only 1 at a time to the output section.
Mixing resistors: will this get the job done?
Imagine a 2-stage Champ preamp as 1 channel; duplicate everything for the other channel. Where the coupling caps would normally connect to the 220kΩ grid reference for the 6V6, have a switch. Center lug goes to the 220kΩ resistor. Outer lug goes to one of the coupling caps, other outer lug goes to the other coupling cap. You get either channel 1 or channel 2.
Mixing resistors... Well, you're problem was that your outputs had parallel paths by which the signal could sneak backwards through another stage's coupling cap and plate load. If you add mixing resistors, you have this possible problem again. I suggested a switch because then you can break the connection backwards through the channel/tube not being auditioned.
The slowed settling you experienced was very likely due to the shared cathode bypass cap and resistor; in fact, if you split those up, you might solve some of your problem. The issue is that when a tube is removed or a new one plugged in, the current through the cathode resistor changes, which changes the voltage across the resistor and the bypass cap takes a finite time to charge to that new voltage.
Anyway, I'm talking about a complete topology change to what you have now, I guess. I thinking Tube 1 = gain stage 1 and 2 of the Champ; Tube 2 = gain stage 1 and 2 of the other channel (that's identical). Input jack can feed both gain stage 1 grids at the same time. Switch between the coupling cap for each channel's stage 2 and the output tube. If you get popping you just can't like with, add a 5-10MΩ resistor from the cap's connection with the switch to ground.
Anyway, this is all far afield from the original intent of the Champ Checker, which was to be a real-life tube tester that offered meaningful readings. It seems like your original incarnation had too many inter-relationships between tubes you were listening to for the purpose you had in mind.