Delighted to hear from you. Your wife and my mother (now sadly deceased) may well be distantly connected. My mother's own name was Bette Bicker and I have been researching her ancestors and their extended families, off and on, for some time. Recently I have been gradually adding this information and supporting documents to my Family Tree on ancestry.co.uk. If you are on this, you should be able to view it as I don't keep it private. My username on ancestry is edwardm_mckeown (I share the subscription with my husband). Otherwise, I can send it to you here or via email, whichever you prefer.
'My' Bickers all trace back to my 3x Great Grandparents, John Bicker (c. 1798 to 1887) and Jane Blair in Newton-on-Ayr. Two of their sons, and latterly John himself, moved to Irvine harbour where they pursued the family trade as fishermen. Another son, a mariner, moved to Troon and this is the John Bicker you mentioned in your original post who was married to Ellen Hay. You thought his parents were John Bicker and Margaret Law. I have realised now that you had identified the right father but Maggie Law was his second wife whom he married in 1853 after the death of Jane Blair.
I can't get beyond John Bicker as his death certificate doesn't give his parents names. There was another family of Bickers in Newton headed by Alexander Bicker, a flesher, who seems to be a contemporary of John. I can't establish if they were related.
My Great Great Grandfather was John's second son, James who lived in Newton till his death in 1880. My Great Grandfather, Mungo Bicker and several of his siblings moved to Glasgow where my grandfather and mother were born. If you can tell me your wife's grandparents' names I should be able to make any connection fairly quickly.
Let me know how you prefer to communicate.
Best wishes
Maureen McKeown