That 1885 birth was recorded is highly questionable; extant Uinta Co. newspaper issues are very few and early on contained almost no "personal" items as well. However, my grandmother was born 1880 "on the ranch" out of Evanston/Almy. Decades later, she had to obtain proof of her age and citizenship, and was able to get a "delayed birth certificate" from Vital Records in Cheyenne by providing six proofs of her birth, i.e. baptism record, school record, marriage record, notarized testimony of older sister/neighbors at time of birth, etc., all with great added information. Your Anderson ancestor might have had to do the same. It's worth a try.
As an added note, the 1910 and 1920 censuses only show two head of households named Beeson who were born in Wyoming (one of them yours). The other fellow might be another issue to pursue. The surname was not common in Wyoming, obviously. Sally M.