Go to a gun shop and buy a proper cleaning rod. One piece, plastic coated. Not a multi-sectioned rod (the joints will scratch up your barrel). You do not want to use a brush on it. A jag or split tip is fine.
6mm = .243cal...airsoft barrels are typically between 6.08-6.03mm...6.10 and 6.01 are the extremes. A .22cal rod will work too.
At most, use light silicone oil to clean your barrel...nothing that will corrode the brass barrel, leave oily residue or harm rubber. Keep patching it out afterwards until it's "dry". It might take a mag of BB's through it afterwards to wipe out any oil left on the rubber...but you'll have to sight in your hopup anyways.
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