Hmm well your not the first or last to be thwarted by 1 silly bolt. Or to discover sometimes 5 times harder to work on something in place. Your doing the hard part- from this end we can only offer suggestions-encouragement along the way..none of us skinned our knuckles once tapping the keyboard.
As I understand your refer-when burner not in cooling mode-gas is throttled back to act as pilot(no separate pilot tube). I suppose if there were restriction thru orifice, when in pilot mode, flame 'could' be just marginally engulfing thermocouple to cause intermittent flame off. Would have been nice to eliminate that possibility, but honestly not high on my suspect list.
If /when you pull & clean orifice/burner & flue down the road, Id replace the thermocouple just out of hand. Because the safety valve is the only thing on that process (pilot) that mechanically 'moves' reacts to power received, assuming its receiving consistent power, its what Id replace. But its outside, accessible could be done after reassembly. The thermostat does control the pilot/main flame gas flow-sending full flow (cooling mode) or thru bypass to limit flow (pilot mode). But I see it either working or not-only as a last resort in desperation would I replace it. It can be tested for gas flow though. My understanding is though it wouldn't be intermittent.
But theres always that everything contributes a tiny bit thing...
My last question out of curiosity, between the 2 manuals, either at Bryants or Lauralhurst (though Bryants shows it) neither give details on the thermocouple, is it removable or easily accessible or is it also 'captured' within the lower box?