Well, many more of ideas were tried now.
I specifically unleveled the feet of the machine until it rocked on 2 points only, no deformation in chasis whatsoever, no effect on beam alignment. Tried to "lift" the left back corner of the machine (where we shim to lessen the vertical walk) - no effect.
I've rechecked all motion components for loose parts, nothing rattles, nothing's loose... Gantry, idlers, rails, gantry-to-rail blocks, etc...
Then, as per many, many, MANY suggestions i straightened the tube to within a millimeter of the tube compartment (51mm from bottom, 19mm from the back wall), then strike mirror1 and go from there... No effect to the problem.
Then we received a video-call from an American Photonics gentleman, who, after receiving all the information confirmed that the machine is twisted. I also was assured that, really, the positioning of the tube doesn't really matter (as i suspected all along, unpopular opinion

) as long as 1st mirror is struck in the center properly. With this in mind he gave me an idea on how to raise the beam without raising the tube higher than the mounts will go. It made a bit more difficult to steer the beam in diagonals, but it is fairly simple, if tricky. None of this helped to alleviate the problem.
In the end i put the machine with ~2mm vertical misalignment back as good as i could with 4mm shimming on TL rail, aligned perfectly at middle of right rail (instead of bottom right as i usually do) to spread the error a bit better throughout Y and left it there.
Now, ALL of the attempts to align the machine ended with the same and CONSISTENT result - TL perfect, BL perfect, BR perfect, TR high. Tube nice, tube crooked, unshimmed (error ~6+mm), all the same result - TR high.
This is proper tube aligned and 1st mirror strike:
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Then there's this - mirror1 holder plate (not the holder itself) mounted off 45deg at the factory. QC my butt, someone just eyed the installation of that. I corrected it before starting the tube position experiments:
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I was thinking of getting the reverse alignment tool from American photonics, but upon thinking on it all it would do in this situation is prove decisively that there is twist in Y, but show up on the left side (instead of right).
At the moment i'm at a complete loss here...