I handload virtually all of my ammo, but I have shot some Winchester Silvertip .44 Spl 200 gr HP rounds through my Redhawk. I find that most of the factory Special loads are very low pressure. Those BH 'Cowboy' loads are probably loaded very light for the CAS events. There are some loadings from the 'boutique' ammo manufacturers (BH?) that are loaded 'hotter' than the typical fodder for modern design/build revolvers.
If the price is right on the Winchester ammo, why not give it a try? For plinking with .44 Spl factory ammo, whatever is the least expensive would be my concern. A lower bullet weight generally gives less recoil, too.
If you do end up hunting with .44 Spl, you should seek out some of those hotter 'boutique' loads.