Went to my local library today and had a look at the book
"Bendigo Advertiser Personal Notices 1854-80"
Their was no reference to the family name Atchinson. (sorry)
The nearest was Atkinson. (nothing close to your William though)
Searching for similar names I came across a James AITCHISON death 15/2/1854 age 21years, Comm Flat. The reference was from a book called "Where they lie, Early Burial on the Bendigo Goldfields 1852-1870 by Annette o'Donohue & Bev. Hanson. Again no reference to Atchinson.
The Pioneer Index for Victoria (which you probably have seen has)
Birth
Annie Louisa ATCHINSON, father William Clark ATCHINSON, mother Elizabeth Porter GILBERT, birth place South Yarr, in 1859. reg no. 12337
Death
Anne ATCHINSON, father William Clar ATCHINSON, mother Elizabeth Porter GILBERT aged 5m, birth place PRA, death in 1859 reg no. 7162
Birth
Gilbert ATCHINSON, father William Clark ATCHINSON, mother Elizabeth GILBERT, birth place Prahran, in 1861 reg no. 19173.
Birth
Laura ATCHINSON, father William Clark ATCHINSON, mother Elizabeth GILBERT, birth place SAND (Sandhurst?) in 1863 reg no. 16943
Death (question)
Elizabeth ATCHINSON, father Clark William ATCHINSON, mother Mary UNKNOWN,age 81, birth place LINC, death in 1869 reg no. 6642.