I think the only way to fix it is to remove it...I might actually have a spare headliner in the rafters above my garage....And I sold my E53 a few months ago...If I do have it I'd give it to you if you came and got it. I am in MA

Removing the headliner is not difficult...I pulled the one I had out of a junkyard car with a pano sunroof. IIIRC, It's only held by 10-20 screws...and was pretty straightforward. It eventually dropped and it slid out the back hatch.

Junkyards rarely sell headliners, so they will give you a good price. Maybe you pull one yourself, get it refinished and install in your E53 all at whatever pace you like.

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'99 E39 528i 5-speed 130k
'06 325Cic auto 115k
'05 X5 3.0 auto 173k-SOLD
'03 530i - manual swapped - 430k miles-SOLD