Thanks for taking up the time to write this great comparison. Very nice impressions on both amps!
I personally think the RCA 6AS7G and 5814 tubes you used on your BHC for the evaluation kinda put the BHC on a huge disadvantageous position. Lots of people find the 6AS7G tube to be sloppy muddy veiled sounding, me included, I personally would put it to the bottom of my BHC power tube ranking. Even a cheap $10 Russian Winged C can easily kick its ass.
If you could unplug one of the Mullard 6080 tubes (which is a great power tube IMHO) from your LF339 or a Bendix 6080, and put it on your BHC, then it would be more like an apple-to-apple comparison. In my estimation of the importance of power vs driver tube on BHC, the power tube accounts for 60% and the driver tube 40%. A sloppy power tube like the RCA 6AS7G will definitely doom the wonderful BHC sounding.
The 5814 tube is also a so so type driver tube, nothing spectacular. If you could use one of the better 12AU7 driver tubes, like Mazda, Siemens, Mullard, or Amperex that
@DenverW or
@GreenNeedle mentioned, things could also turn better.
Now, I am not too familiar with the EF86 tube and how it is used in LF339. It is a pentode, so I would guess most likely it is used as a triode-strapped pentode in LF339, since we don't need a lot of power from pentodes in the driver stage, and linearity is the key in this stage. I would suggest using the same power tube (Mullard 6080 or Bendix 6080) on both amps, and using the driver tube from the same brand (say, Mullard, Sylvania, RCA) for a fair apple-to-apple evaluation.