I didn't use the keg: it seems it's there only to force you to take the staff on the token (but that doesn't change the whole puzzle, I think)
What you need to do is quite clear: reach the conquer token without laying any egg on the plate, maneuvering yourself so your back is never next to a pressure plate.
Some hints
I started
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×Moving SE facing SE
and then
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×reach check point at 16,19 - you can go there using a "short path", no need to take a long road through the vertical corridors
then
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×take a short path to 26,18
then....
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×reach 28,8
finally
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×go south to the conquer token
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×I had to turn my orientation before moving south