You mentioned that you looked for 1850 - William Bishop.
I am actually looking for a William Bishop in 1775.
You started off by saying that you were looking for a death notice. There is a small chance of finding a death notice around 1850-ish, which is why I looked for a death notice in 1850.
You have William Bishop, married to Helen Maxwell, probably daughter of Alexander Maxwell, in Canongate on 1 October 1799. The index at
www.scotlandspeople.gov.uk lists three children:
David, 20 January 1803, Edinburgh
Margaret, born 7 April 1805, Edinburgh
Alexander, 7 June 1807, South Leith
John, 29 March 1809, South Leith
Helen, 12 December 1813, South Leith
There is room for a child before David, and one more between John and Helen. This is inconvenient, because it means that you cannot assume that David was the eldest son, so you can't really apply the naming tradition.
If they were married in 1799, William cannot have been born after 1785 (and was almost certainly born before 1780). Therefore he must have been at least 45 in the 1841 census.
There are only five William Bishops aged 45 or more in the 1841 census
One aged 86 in Livingston
One aged 66 in South Leith
One aged 54 in West Calder
One aged 50 in Livingston
One aged 50 in Whitburn
The one in West Calder can be discounted because he is still in the 1851 census, aged 66 (so born 1784/5), unmarried, born Kirknewton, with sister Catherine, also unmarried.
The William Bishop, aged 66, chimney sweep (so born 1774/5 if his age is correct), with a family consisting of Catherine, 45; Mary, 20: Abraham, 16; Anne, 10; and William, 5 must be the one who married Catherine Ross in 1823 and had Abraham Connell, 15 December 1824 and Anne, 1 June 1829. But who is Mary, listed as 20, so born 1816/1821?
(Catherine, who has got 20 years older in the 10 years, is still at 2 Quality Street in the 1851 census with daughter Ann and grandson William. She was born in Logie, Ross and Cromarty.)
I notice that you haven't actually said that this is your William Bishop but there doesn't seem to be a more likely candidate. Is he the widower of Helen Maxwell, married after her death to a second wife? Could Mary be a younger child of Helen Maxwell? Have you found the death of Helen Maxwell?
If this is the William Bishop you are looking it, notice that the 1841 census says that he was not born in Midlothian. This means that whoever he was, he cannot be the one previously suggested, son of James B and Janet Christie.
There are six listings in the IGI of William Bishops born between 1770 and 1780 in Scotland but not in Midlothian.
12 December 1773, Whitburn, West Lothian, William Bishop and Agnes Weir
7 June 1774, Kirknewton and West Calder, West Lothian, James Bishop and Margaret Flint
3 December 1774, Falkirk, Stirlingshire, William Bishop and Margaret Flint
7 April 1775, Kelso, Roxburghshire, William Bishop and Jane Hood
but if you believe the IGI (risky!) the one born in Falkirk married Janet Leishman in 1799.
So unless there were other William Bishops whose births/baptisms are not recorded (perfectly likely) your one could be one of the remaining three, which at least narrows it down to three.
