Cleaning the wrap without a lathe is not easy. Also a lot of older wraps are starting to rot and fall apart when repressed. You can take it and let your repair man try to clean and repress it and if it does not work replace it.
I am not a cuemaker but I have had excellent success carefully cleaning original linen wraps from the 1960's.
I use a damp, not wet, cloth or paper towel with mild detergent and immediately dry with a dry towel. I clean roughly 4" at a time and dry it, eventually working my way up the wrap.
I do this by hand without a lathe.
That looks like a nice job on the cleaning. How do you re-press and polish the linen without a lathe?
To render my linen wraps smooth again, I burnish the wrap using a round smooth glass object........the flax strands woven into the Cortland Linen lay perfectly flat and smooth afterwards.......the wrap actually acquires a sheen.