I would stitch together and then process.
But you can batch process for the some standard type processing in Lightroom. Things like a tweak to white balance, some noise reduction, increase saturation a little.
So long as each panel gets the same treatment you should be fine. Then final colour processing at the end.
A lot of the noise may be reduced by the large overlap and also so many panels will mean the noise will go to the background I would think.
Greg.
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