I wouldn't order the certificate just yet - you need to weigh matters up.
If you look at Carlisle births, there is one for September 1876 and one for March 1877 - just Mary but there are several for Mary ......
There is also a death March 1877 for Mary aged 0 which could be either of these but probably the 2nd one.
Similarly, there are several marriages for David Graham 1871-1875.

I keep being drawn to a family David and Mary Graham in Burgh with several children. In 1851 they have a son David b 1843 but then he disappears. Again about 4 deaths 1851-1861 so as we already know this is not an easy task!
I was wondering if Mary could be a grandaughter of this family through an illegitimate birth to a daughter perhaps and gives David as her father's name- nothing to indicate this though.
If you have access to Carlisle records, that would be great Bob.
Added: last two posts were made whilst I was doing this but I will leave it anyway.