K&T has some slight peculiarities, but it's still wiring.
The less old K&T has been messed with the less it is a worry. It's kinda like an old truck, most of the bad ones break down or get wrecked. The ones that are still on the road survived for a reason.
Pithy, and an accurate analogy. I like it!
It sounds like the OP situation has the hallmarks of a real honest to gosh challenging troubleshoot. These are the best test of one's skills.
- Is the K&T added after original construction, or was it roughed in as part of the original construction? This will help you make generalizations about its routing in relationship to other conductive metallic systems such as metal piping or ducting.
- What are the grounded systems in the house? If the dwelling is wood framing, plaster and lathe, wood exterior, and it hasn't been significantly altered over the decades, these are likely to be only gas, plumbing and heating runs. The incidental or deliberate grounding of these is part of the troubleshooting landscape.
- Since the OP has already spent some time exploring openings with no success, the mapping of the affected branch circuit should be considered. It is a rare original K&T circuit that is still pristine, most have extensions. In early K&T extensions, armored cable was commonly used, and the armor did not have to be grounded. When examining openings, make note of whether the K&T conductors enter the boxes through individual nonmetallic loom sleeves or enter at a locknut fitting, which may be either armored cable or rigid metal conduit. Pay attention to the presence of the armored cable and test for grounding. The armored cable was routinely fished in and could be simply laying in contact with the grounded systems in #2 above.
- Original K&T circuits get trimmed back, in dwellings that have a history of electrical upgrades, and, being lightly loaded, the K&T gets spliced, at a junction box, together with other wiring methods. Historically, K&T provided "smoke-less illumination" which was the "high tech" wonder of the late 1800s and early 1900s. Historically, there were VERY few receptacle outlets in the really early stuff. So pay attention to the load connected to any, and all, of the receptacles of the complete branch circuit. A plugged in load may well be the fault.
- Unplug every thing, and take all the light bulbs out. If smoke detection is extended off the K&T, disconnect it. You should be able to "float" the neutral and energized conductors. Find the feed end of the K&T and disconnect both the neutral and hot conductors. With all load removed, there should be no continuity between each other and between each to ground.
Some of the interesting shorts I have found have been the heat-aged-and-deteriorated metal chandelier reversed polarity lampholders that are shorting to gas-pipe-hanger grounding. Another was the K&T installed as part of the original construction of the building where a steam heat pipe was installed after the electricians, and the pipe was installed in contact with the K&T conductor (the pipe fitters didn't inform the electricians, the required "tube" protective sleeve was not installed, and, I found, 95 years later, the heat degradation of the insulation and its crumbling had resulted in a spot weld of the copper conductor to the steam pipe.)
The most common cause, though, is the cord and plug connected load, at a receptacle that is in an area that doesn't "seem" to be part of the area served by what one "thinks" is where the branch circuit should be.