As a member of a casting club that invites and coaches new flyfishers, I see this problem often. I don't agree with adding weight inside the butt of the rod - only for the reason that it matches the rod to only one reel (or one of equal weight) and the weights are hard to remove. Our club suggests adding weight to the reel, which allows for changes in the future (changing the reel or eventually using that reel on a different rod).
I like the lead core line approach, although we use strips of lead the width of the arbor of the reel wrapped in the center of the reel. You can also use heavy split shot attached to the backing near the center of the arbor (spaced evenly around it).
With most of these weights, it is possible to hang or tape them to the rod butt to experiment. I suggest you do this with about 30’ of line outside the rod tip to properly weight the system as it would be for the majority of your fishing.