Oh, boy, RR man, that's why you're probably having a problem. They often picked up and traveled to other places by train, sometimes taking family and all.
I don't see any Beauchamps in Angelina Co, TX for the 1910 census, before birth of Homer. There's a Stephen in 1920 about the right age to be a brother of your William Henry, but on Rootsweb's WorldConnect someone has done some research on this Alexander Stephen Beauchamp, and if they are correct he does have a half brother named William H., but he was born in the 1850s.
On SSDI there is a "close" match.
Beauchamp, William, b 2 Oct 1877; d May 1972; last residence Stillwater, Payne Co, OK; 94 years old; SS card issued in AR.