If they survive, you should be looking for the 'day books' which the parish clerks kept for such things.
I've seen the ones at the London Metropoplitran Archives for St Marylebone church and I do believe some are online with Ancestry.
However, whilst the ones I've seen show how much was paid for the funeral, they don't always specify where in the churchyard someone was buried. There has been a church in the area for 1000 years and a lot of people have gone into the ground during that time. While the workhouse was in existance the churchyard was used for 'pauper', burials and they just went in one after another.
St Marylebone, at one time, had a huge churchyrad, now redeveloped into a garden and partially built upon by a girls school.
Sad indeed but a sign of the times when the church no longer feels it can keep up the maintenance and it gets built on.
I had a look at the archives web site and the catalogue for the church's holdings is here
http://calmview.birmingham.gov.uk/CalmView/docs/EP%2035.pdfhave a look down it to see if there are references to account books etc.
There is also a reference on the last page to 4 volumes of grave locations that starts in 1833. It says they are still at the church but may have been deposited later and the page not updated.
Contact the library again and ask them what the situation is with these records and ask the same question to the church. Maybe the current incumbent doesn't know of their existence.
Dawn