I think those sharpie pens are alcohol based. if the cue has shellac or lacquer it may bother it, but unless it's old it's probably got some new style finish like epoxy or UV cured coating and that may be somewhat resistant to alcohol.
the 3m polishing compounds may work otherwise, they come in different grades and work well for bringing old plastic items back from being oxidised.
look for plastic polish, or maybe a cut polish for a car, that wouldn't be so harsh. maybe you have something you use to clean pool balls and such? turtle wax? silver or brass polish? they sell kits to fix fogged headlight lenses and stuff. Id see what you have around the house. toothpaste may be a bit coarse.
sometimes I lacquer antiques and use gloss, but I don't really want a bright shiny gloss so I buy a product called "wool lube" from mohawk finihshing supplies, you can try just a hand soap or leather cleaner or similar. it only takes a dab.
I use fine steel wool and rub the finish out, this removes that high shine which does not belong on an antique unless you want it to look like it just left the factory. it basically makes the fine steel wool not fall apart and it cuts better , like wet sanding , sort of. im left with a shine but not like a new piano, I want a smooth rubbed finish like an old worn antique hand rail basically.
I like that way because if I spray lacquer and don't have a perfectly dust proof area it hides the sins or dust nibs, cause by some amount of dust. some use burlap and pumice powder, similar idea.