If there is enough metal outside the present bore, I would make a bush, turning the OD and the bore without disturbing the work inn the chuck. In this way, OD and bore will be concentric.
Choose the OD to keep a sensible wall thickness for both e bush and the boss of the pulley.
Bush length should be same as that of the boss of the pulley.
Measure the OD that you have produced.
Now e get into the Catch 22 land if the pulley is part of the drive to the lathe!
The pulley needs to set in the chuck, or under a milling machine so that the rim of the pulley is square to the required axis. Bore the pulley.
If you are going to Loctite the bush into the pulley, bore it no more than 0.003" (0.075 mm) larger than the OD of the bush.
If the intent is to make the bush an interference fit in the pulley, bore the pulley to 0.0005" SMALLER than the OD of the bush.
If the pulley is alloy, don't overdo the interference, for fear of cracking the pulley!
The bush goes into the freezer overnight. Before taking the bush out of the freezer, put the pulley into a pan of water and boil it for at least 10 minutes.
You may want to make up a simple puller, as a "Just in case". A length of studding (M8? ) and two large washers and nuts.
Then remove the bush from the freezer, and then the pulley from the hot water and press the two together. If you are lucky, the bush will just slide into the pulley. If there is any resistance quickly thread the studding through the bush and pulley, Have spanners to hand. (Fit one washer and nut to studding, in readiness first! ) then fit the second washer and tighten the nuts to pull the bush into the pulley.
If the pulley bis locked to the shaft with a grubscrew, you will need to use the existing tapping as a guide to drill and tap the bush.
Hopefully, by now you will have a pulley that runs true, and is ready to refit.
Howard