Depending on what tooling you have available could change how you do it.

Flat crowning is a option and the easiest.

Lathe would be ideal, but could be a pain considering depending on how it was drilled before as it might not be centered. A drill press might work better if they drilled it off center as at least you could pick up the hole easier. I am not sure I would buy a special drill myself, I likely would just cut what is required with a standard high speed drill with a 140 degree point on it (hard metal angle), taking time to grind it carefully. Then I would grind up or have a flat drill finish it off once I felt I cleaned up the damaged area. As to drill size, a 6.5 Arisakaicon (roughly .264" bore) should be able to be cleaned up nicely with a 9/32" (.281") drill if its nicely centered (large enough that it shall erase all the rifling well not taking too much meat out of the barrel). If the damage is more extensive than that, you can always choose a larger drill to compensate, but 11/16" seems excessive to me.

Note I have never counterbored a barrel, just basing this off the machining I have done and continue to do for a living.