this is, folks, a fictional story. Fiction writers write fiction, meaning they make stuff up. And sometimes the stuff they make up does not track perfectly as 'realistic'. It's entertainment, people!
I agree that Don's switcheroo on the battle field was a drastic way to escape Dick Whitman. . . I think the reason Don's secret is such a burden to him is because his real secret, switching identities on the battlefield was a cowardly act that is likely to be scorned by most people.
Dick tells Don, when they meet in Korea, that he has only been in the Army a short while and he is grand new to Korea. Heck, Dick probably enlisted expecting to be stationed state-side, right? He didn't realize that as a poor, unconnected grunt, he would be shipped straight to the front?!!
Dick switched those dog tags because he had a LONG time left in combat position: I don't know how long guys were sent to Korea .. but I think the minimum amount of time you could enlist was two years. . . so give two months for basic training. . . wouldn't that mean Dick was facing, easily, another year in Korea as a PRIVATE, meaning he was going to be digging ditches when he was lucky and in direct combat when he was unlucky. I saw him digging that ditch, and watching Don do absolutely nothing. . . remember how Don told Dick to start digging, that he was the only one. Don made it very clear that he wasn't going to do that work because he was superior to Dick in rank. Don flaunted his rank. I think that with each shovel full of dirt, Dick's resentment of Don's privilege surge. He had to be thinking about why Don got to be an officer and he a lowly private and what was so special about Don and his elite ability to just sit around and watch one guy dig? I think all of Dick's class resentment had time to fester as he dug, even if it was just a day or two . . . I think he had spent a whole lot of thought wondering what it would be like to go through Korea as an officer.
It was convenient that Don's hitch in the army was almost over -- he only had a few more weeks. . Dick switched dog tags as a way of deserting.
Don's big secret is the cowardly act of desertion, not the name switch. Living with an assumed name has given him a way to dissociate from the underlying cowardice. . he can snivel and cower about his fear of being found out but his real fear is having guys like Roger and Bert Cooper and even a sniveling drip like Pete Campbell found out that in the battlefield, in combat, he took a cowardly way out.
Now don't get me wrong. I don't want any humans to be trapped into war. Desertion is okay with me. I think there should be no military. Let's all turn the other cheek and trust that the intrinsic goodness that is commonly labeled 'god' or 'allah' will carry us through.But our society, Don/Dick's society, attached a lot of scorn on deserters. Our military complex needs that scorn to perpetuate itself. If everyone shirked their combat position the way Dick did, how could the guys on the top keep practicing war all the time, war rooted in profiteering?!! In the military-industrial complex, there aren't many crimes greater than desertion. . . and that is the root of Don/Dick's secret.
The name change, as has been suggested, is not that big a deal.