Man, what a battle.
I think you are on the right track with the battery.
Low voltage will absolutely affect the shifting strategy.
You've checked the sealing sleeves as well I'm sure.
If not I would replace them but I think your issues are voltage related.

Sent from my Pixel 4 XL using Tapatalk

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04 E53 4.8
98 E38 4.4
04 Touareg 4.2 (x2)
87 E30 (getting a 4.4/6spd swap)
89 E30, selling it
96 E36 parts car
MK2 jetta with a 1.9alh swap
90 XJ Crawler
2000 XJ crawler in the works
63 Ragtop bug
64 Bug
06 F-350 to drag all the above when they brake.