In the maintenance order, according to the transcript, the mother was Mary
Barnatt, not Barratt.
And, yes, you do need to see the maintenance order, which is why I said that you need to contact Lincolnshire FHS to find out the database the transcription is in. Then you have a chance of finding the reference number if you need to order a copy of the document from Lincoln Archives. If you can visit the Archives then they should be able to trace the document simply by the place and year, and then the names. (Archives might not need the document reference if you order by post, but it's always better to quote one.)
On Elizabeth's death certificate does it give John's occupation at all? Or is the address Southgate, which is where they were living in the 1841 census?
I would say that it is highly unlikely that Mary Barnatt and Elizabeth Barrand are the same person. It's more likely a simple case of enumerator error in the 1841 census entry.
From what I can see the children's mother is always named as Elizabeth when they are baptised. No children listed with Mary Brand as mother between 1821 and 1841 in Sleaford St Denys baptisms.
FreeREG will be very useful in helping you to find the dates of the baptisms.
http://freereg.rootsweb.com/Search for baptisms, John Brand, 1818 +/- 10 years, in Sleaford, Lincolnshire. It's not as silly as it might sound, because the results will contain not only children called John Brand but also children whose father is called John Brand. One word of warning is to always check the father's occupation if given, as there can be more than person with the same name having children baptised in that church.
If you check the burials (best to search on the name of the child) you will find two of the children died as infants.
Sleaford baptisms 1743 - 1812 are in this register (as well as marriages 1743 -1754.)
http://www.lincstothepast.com/Baptisms-and-marriages-/814202.record?pt=TBaptisms for 1786 start on image 40; 1809 start on image 65.
If you go back a few images, you might find an entry for Mary Barnatt's child being baptised circa 1806.
Sleaford baptisms 1813-1836
http://www.lincstothepast.com/Baptisms/513556.record?pt=TYou might also like to know that a Mary Bar
nett married a Robert Cobb in Sleaford in 1807. (Reference from LFHS Lafford deanery marriage index 1700-1813. Index now re-issued on a CD containing 4 adjacent deaneries and more detail from the PR.)
http://www.lincstothepast.com/Records/RecordDisplayTranscript.aspx?oid=628744&iid=324152The trick with searching for PRs on LLTP is to read the help section first, especially the 'how to search for parish registers'. Read to the end of the document, because it gives a list of the parish names to input. Then in the 'with the exact phrase' box on the 'Advanced search' page enter the parish name ending with PAR 1, e.g. for Sleaford enter Sleaford PAR 1. Just be careful because you do sometimes get 'rogue' results included. In the case of Sleaford, for some reason you also get Asgarby by Sleaford results as well, but they are clearly marked against the 'reference'.
PRs prior to 1813 are often called 'general registers', and depending on the years covered can contain baptisms, marriages, and burials in any combination.

Sometimes there is more than one register covering the same or similar dates so it's worth looking through the complete results list.