I hunt with my 1727 pattern Tulle Fusil de Chasse but its .62 bore is much smaller than the .77 tube on the Bess. I expect the Bess will work about the same as the Tulle; let the deer get close enough to put a ball through its lungs, press the trigger carefully, get out your knife.
I have used a Southern Appalacian smooth rifle with 30" tube. Load was .600 ball patched with .020 oxyoke. Ball is ~325 grs and powder is 80 grains of GOEX 2 ffg. English dover flint knapped sharp. Distance 50 yards through the lungs stopped on off side. Lunged maybe 25 yards and collapsed.
I wouldn't expect much different results from a Brown Bess.