Here's the label of the original UK release of the Love Me Tender EP.
You will notice that writing credits go to "Presley - Matson" with not a mention of the man who actually wrote the music, George Poulton.
Fair enough that Fosdick's name is absent: he provided the Aura Lea lyrics, but had nothing to do with those of Love Me Tender. It was quite a different matter with Poulton, however.
Elvis's first film, made in 1956, was to be a Civil War era western called The Reno Brothers. Elvis was as Tom Parker did (Tom Parker was his infamous manager, an honorary Colonel and an illegal Dutch immigrant), so rather than being given a purely dramatic role, Elvis was made to sing four songs in the film, three of which were quite inappropriate for the time in which the story was set (Let Me, Poor Boy, and We're Gonna Move). The fourth was at least to make use of a contemporary melody, though with a new set of lyrics, provided by Ken Darby, the film's musical director (the other three numbers were also the product of Darby's hand). Darby used the Civil War era tune of Aura Lea, but substituted Fosdick's original lyrics for his own, those of Love Me Tender. For whatever reason, Darby did not want his own name shown on the credits (nor, it would seem, did he want Pouton's name to appear), so he assigned those credits instead to his wife, Vera Matson, and (no doubt at the "suggestion" of Parker) to Elvis himself. Darby said that he gave his wife half the credit because, "She didn't write it either."
20th Century Fox, the film's producers, refused to allow Elvis's usual partners, Scotty Moore and Bill Black, to accompany him, asserting that they were unable to play country music(!). Instead the Ken Darby Trio provided the accompaniment. To make matters worse, rather than using a regular recording studio, recording took place on a large film soundstage. On 24 August 1956, Elvis recorded three of the tracks required for the film, including Love Me Tender (Let Me was recorded on 4 September). In fact, two versions of Love Me Tender were made, the end title version, using slightly different lyrics, was recorded on 1 October. You will be familiar with the "normal" version, so here's the end title version: