What is the best sounding Mahler symphony cycle?

I think that you are being hard on the Budapest Festival Orchestra.  I am not claiming that it can equal Chicago or Vienna , etc.,for lung power, and I thought I made that pretty clear.  What they have done, however, remarkably well, is create a MittelEuropa type of sound, that may have been what the finest Orchestras that weren’t Berlin or Vienna may  have sounded like in Mahler’s day (and let’s not forget that GM worked quite a bit in the Hungarian Capital).
  If you wish to limit your listening to “how the very  top Orchestras perform these works” be my guest.  To imply that the Budapest is some kind of semi pro outfit that is out of their depth, however, is engaging in hyperbole.  I agree that the Fisher /Budapest Ninth isn’t competitive, as a performance with the very best, nor have I heard any Mahler recording by them that displaces any of my favorites.
This isn’t because the Budapest cycle is bad, but because the bar is so incredibly high.  Look how many Mahler recordings are out there!  Most of the greatest Conductors of the last 50 years are represented, along with quite a few “Who?”  What the Channel Classic recordings, are, however, are the most realistic reproduction of how the actual Orchestra sounds that I am personally familiar with.  They represent a fine Orchestra faithfully reproduced.  The performances are at worst passable.

Why stick to one conductor and one orchestra? You can compile your own favorite cycle.
Here’s mine:
#1: Bruno Walter / Columbia Symphony (columbia USA)
#2: Otto Klemperer / Philharmonia Orchestra (emi/columbia UK)
#3: Zubin Mehta / LAPO (decca)
#4: Georg Szell / Cleveland Orchestra (columbia UK)
#5: John Barbirolli / New Philharmonia Orchestra (hmv)
#6: Georg Solti / Chicago Symphony (decca)
#7: Leonard Bernstein / NYPO (DGG)
#8: Georg Solti / Chicago Symphony (decca)
#9: Bruno Walter / Columbia Symphony (columbia USA)

This is almost exclusively based on the original vinyl issues (except the Bernstein 7th), but I’m sure these performances are all available on cd or SACD. Don’t know about the quality of the digital transfers, but the sound quality on vinyl is mostly excellent (exceptional in the case of the Mehta 3rd).

@edgewear

Of course no one is restricted to just one cycle, and not every conductor has an equal affinity for each piece.  So here goes.
#1 Horenstein/LSO
#2 Abbado/Chicago 
#3-Bernstein/NYP first recording. Abbado/Lucerne 
#4-Szell
#5-Honeck/Pittsburgh or Walter/NYP (mono)
#6-Karajan/BPO. #7 Bernstein/NY Phil (Sony)
#8 Solti
#9 Karel Ancerl/Czech PO

Das Lied-Reiner or Walter (stereo)

#10. Without getting into the question of which edition—Dausgaard/Seattle 

I like your choices, all of which I have in my collection, except the Mehta (had it on  vinyl but sold off my analog rig a few years ago).

Re:Budapest. I didn’t say they’re terrible. Just way way overrated. 
Good choices above from both of you. The Mehta LAP Mahler 3 for me is a bit polite. Very well done played but the recording itself strikes me as being from just a bit too far away. It looks like the preferences for both of you skew to older recordings perhaps hints at your generation. My preference skews towards many of the newer recordings from the big 5 plus LA with notable exceptions for Chicago and LA from the mid 20th. 
I have many of these on vinyl as well (not the most recent ones of course which aren’t on vinyl)

It seems both of you like the Bernstein #7. An excellent performer, but I never felt the brass in the NYP from that era up to the task of the very brass-heavy 7. I’d recommend the Cleveland (Boulez - 1996) recording. It’s simply stunning. 

My choices have to do with vinyl being my primary source, but I guess they date me as well 😕

On original vinyl the Mehta #3 has no rivals sonically, although as a performance the Bernstein (both the CBS and DGG recordings) may be superior.

I forgot to mention Das Lied von der Erde, but here Klemperer on HMV Angel has my preference (yep, another oldie). As for #10 (Cooke version), I like the Rattle on EMI, but I'll admit that I haven't heard any other performances.