As I mentioned before, the PRs for Brockford would come under the PRs for the church of All Saints in Brockford (assuming the Sullivan's are CofE - although they might well be RC).
I have found an Anna Adelaide Catherine Sullivan in the Sudbury Union Workhouse (now Walnuttree Hospital) - in 1871 she was age 28 years and gives her birthplace as Wetheringsett.
I noticed a baby born in 1909 Edith Mary Sullivan in Stow District (died 1910).
The Middy Railway was one of the last to be built - and they employed Irish workers. I wonder if the family were in Wetheringsett/Brockford for a short while?
This sounds rather like my own line of Irish named ancestors - I have two great grandfathers - one paternal one maternal. Both were from Cork, but one (Clancy) insisted on all Census returns that he was born in St Marylebone and the other (Wilson) went off to the USA to make his fortune and although he sent for his wife and three daughters, she would not leave the UK.
I have also found that the use of middle name as the family name is common in my Irish connections.
Supper time, better cook now

Pat ...