I hunted blacktails in California for about 15 years and helped dress deer that were harvested with 243, 30/30, 270, 30-'06 and even a 30 caliber M1 carbine. They are generally lighter and even more fragile than the mule deer and whitetails most of us hunt...they are not difficult to kill. Frankly, the 270WSM, 30-'06 and 300WSM are all too much gun for these little deer, IMHO. The best option, for someone who does not reload, would be the 30-'06 with 150gr loads and either cup n' core bullets or one of the bonded options, like the Accubond, if it can be found in a factory load.
To avoid bloodshot meat you really want a heavier bullet that will still open up, but not violently. I found a standard 270 with 140gr bullet, loaded on the mild side, killed deer faster and resulted in less bloodshot tissue, should your shot wind up in the shoulder instead of back in the ribs. My uncle used an '06 and was very frustrated by how poorly his 180gr bullets killed. The deer hit with that big slug had a .308 hole in and a .310 hole going out, with the deer showing little or no sign of being hit. When he switched to 150gr Core-lokt bullets he was much happier with the end result. My dad shot several deer with his .243 and they usually fell in their tracks or within just a few yards. He kept the bullets well back from the shoulder so there was no meat damaged. Out to 150 yards or so, the 30/30 with 170gr FP did the best job and ruined very little meat, but it would never give you those 350-400 yard shots.