My land is surrounded by another man's property. My neighbor has introduced fallow deer and they spend a fair bit of time within sight of my house. The fallow deer are only slightly larger than the local blacktails, not nearly as big as whitetails. Haven't shot a fallow deer, but would have full confidence in the 250 grain Winchester Partition Gold my Redhawk likes. I wouldn't hesitate to use any premium bullet, though I might shy away from those under 240 grains in weight, not for terminal ballistics, but because my .44 is generally more accurate with the heavier bullets.
Fallow deer are European imports that are the favored estate deer. They are the result of centuries of management and are in a fuzzy zone between wild and domestic. Run a search engine on fallow deer and you will get plenty of hits on their biology, size range, etc.
If you haven't seen fallow deer yet, be prepared-the majority of them around here are white(very spooky in the late night fog I first saw them in), a few are the palest blonde, and only a very small minority are red/brown. The dark coated deer keep their spots into adulthood.