Last thing I would want in a p/dog patch would be a 243...let alone an 243 AI
Even a heavy barrel....noise/recoil/heat.....
Many better 6mm choices as even you mentioned
6BR non repeater....30-31 grs....my ole fav the 6XC...37-38 grs
Oh....I don't buy store bought ammo so I don't worry about running out