I called Pedersoli about this and they told me a 500 gr or heavier bullet would give me the best results. IMO 500 is overkill. Shooting at a target up at our cabin using a 500 gr cast bullet the bullet went completely through a 10" hard maple and buried itself in one behind it deep enough that I couldn't get it out! So a minimum of 12" of penetration in a green hard maple is enough to pass through just about any living creature except maybe an elephant's skull.
Frankly though I don't believe Pedersoli was giving me the correct answer to my question. Reason being that the Pedersoli has a 1:18 and my Marlin 1895s have a 1:18 twist. The Marlins shoot great with 300 gr jacketed bullets. I believe the 340s should be good in both.