You’re right, it’s not a target rifle but a military rifle intended to be used with Minie bullets. They work with shallow rifling - I shoot an original .577 Enfield that has progressive depth rifling (it gets shallower from breech to muzzle). So far this year I have won a 200 yard offhand match and a 500 yard prone match with it - with load development they can be used for target shooting. It’s not altogether clear from your posts what bullets you are using, but changing to a ‘tighter deep rifled barrel’ may not bring you the results you seek. It may also limit you to patched round ball and the relatively short distances that ball will remain accurate for.
David