Clevo...
my best recollection seemed to point to a magnetization problem in the head/transport and/or the tape. It's more of a "static" pop and click, and is very random, though is regular and persistent, too. I know that's a contradiction in terms! To me, it seemed to be generated in the cassette shell as the tape unspooled.
I've had this problem once before, so I've heard it and I think I know exactly the sound you're hearing. However, at the time, to remedy this problem I did not demagnetize, but I switched to another similar and unused/idle Portastudio to finish the session. Not sure if I went from a 424mkII to a mkIII or vice-versa. I believed at the time, and to this day, that it's a static and/or magnetization problem. It is a static pop, is what's cemented in my mind.
It strikes me that if you demagnetized once, it may come back and have to be demagnetized again. To me, that would make sense somewhat. Static depends a lot on what part of the country, what weather conditions, what your power condition is, and even the interior static tendencies of your household furnishings, decor, carpet, etc. I'm in the high desert, so we have low humidity and static as a concern most days of the year. A factor would also be the grounding of your power source,... though I know the PSP-424mKIII is a 2-prong jobbie, but if your work table and studio area were static proned,... (rubber or plastic mats, formica table tops, etc), then this problem could be locale specific, and/or made worse by certain household/studio/lifestyle issues of static charge. Sorry to make the scope of this problem wider, not narrower, but such is the universal nature of static. You may already know that the sheer act of separating two pieces of mylar generates plenty of static, eh? (Like the unspooling of tape is peeling mylar off the spool). You gotta know that that process inherently generates a TON of static, and perhaps could be worsened or not-nullified by a high static-prone environment.
Sorry I don't have a more concise solution, or 'silver bullet', but I can relate to your frustration under the circumstances. The 424mkIII truly is a leading quality machine, and it's worth rescuing.
Tascam Service in Montebello has been flawless for me in the past, but as your problem may be a bit more esoteric, it would perhaps be difficult to troubleshoot 'in the field', hence, shipping your unit to California may change the testing constants, somewhat. I also believe that the cassette shell can carry and propagate this charge, and so testing at TEAC with a "known good tape" might not have shown up the problem, at all. Sorry for the run-on sentences!!!
Look again at magnetism and static, both of the deck and the tape. Perhaps for a blank tape it would not hurt to bulk erase and demag the tape completely before testing the 424mkIII and/or using it in a session.
Best of luck to you!!
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