I can testify to the accuracy of round balls out of the Deerstalker. 80 grains of 3F powder, 0.020 patch with a .490 round ball. That gun will have no trouble putting them in at 50 yards, and should do well out to 100 yards. Farthest I ever shot at game was a groundhog at right around 100 yards. I can't say much about the peep sight; I find them harder to use than iron sights, but, and no offense is meant, my eyes are probably much younger than yours (this is an assumption; I of course could be wrong). I know my dad is starting to prefer a peep to irons now, so it may very well help you. Good luck!
A bad day hunting rabbits beats a good day of anything else!