Elvis wasn't just a reflection of his times
People need to use their imaginations when they view a movie like LMT, and look at it through the perspective of being there at the time, without prejudice of hindsight. When the idea of the songs being performed by Elvis's character in 1865 was put forth, the writers should have, and maybe did, have in mind what a guy like Elvis might have brought to the culture of that era. Yes, there were probably no screaming audiences or wild hip gyrations back then (at least among whites), but there weren't in 1954 either; then Elvis came along and changed everything. Who's to say he couldn't have revolutionized entertainment at other periods of history? Certainly he was influenced by 20th century music genres in his pioneering of rock and roll, but that was taken into account in the movie, with the songs being ballads and old-fashioned country-western in style, without signs of the rhythm and blues he was accustomed to (he even had to play without his rockabilly backups for the first time). If you are, like Elvis, a guy with a great sense of rhythm and ability to dance with originality, you would have been a potential icon no matter when you were born.
Brimming with youthful energy and excitement was what Clint Reno was all about, and it follows that he would go about singing and dancing in the same way, even if it was a novel approach. The only time he ever slowed down in the movie was when he sang the title song to his elderly mother. Those who may think Elvis overacted, or was out of place on a stage in the 1860's need to look at the character as a whole, and how Clint paralleled a young Elvis Presley just reaching acceptance as an adult and as an entertainer.
BTW, too many people also do not appreciate the Elvis of '56 because of foreknowledge of how he ended up, like the caricature presented by today's Elvis imitators. Once again, put yourself in the place and time of the original release and you can see history much clearer.