Take that egg sinker and lay it sideways on an anvil of some sort and lightly tap it while rotating to reduce the dia. to a bit oversize for the bore.
You dont need to heat up the barrel, but comfy, inside room temp would be nice.
Lube the barrel up good
Tap the sinker into the bore with a SOFT faced object (a plastic mallet or a hardwood board), then drive it deeper with a almost bore dia hardwood dowel (to keep from messing up the bore with a metal rod slipping off the slug and to the side into the rifling)
Once it take the bore it will be easier to drift through
Measure, make a note, then put both into a small RX bottle (or such) for future reference