Elizabeth Paterson married Robert Gillespie in St Ninians in 1848.
In 1851 Robert Gillespie, 29, coal and iron miner, born Tillicoultry, is with wife Elizabeth, 24; son David, 2; daughter Elizabeth, 9 months, all born St Ninians. Note that she named he first daughter Elizabeth, not Catherine (and there is no sign that she had a daughter named Catherine at all). Also that there is a record of a Robert Gillespie, parents David G and Agnes G, being baptised in Clickmannan in 1821, and that Robert's eldest son was named David and her second daughter Agnes.
Neither David nor Elizabeth Jr is in the IGI, but there are two younger children, Agnes 1857 and David 1863, born born in Denny.
I don't have free access to the 1861 census, but the index on FamilySearch lists in Denny Robert Gallespie, 40, born Clackmannanshire, Elizabeth, 33; David, 12; Elizabeth, 10; Robert, 7 and Agnes, 3. Unfortunately the index only lists the birthplaces as Stirlingshire, but the original would specify a parish.
Likewise she is listed in 1871 as aged 43, and still living in Denny, as is Robert, aged 49. I cannot access the 1871 census free either, and I can't find any of them at all in the LDS CD-ROM of the 1881 census.
From her age in the censuses, she would have been born 1826-1828.
The (community contributed: beware!) IGI lists 5 children of James Paterson and Catherine Ferguson: Fanny 1824; Elizabeth 1826; James 1828; John 1831; William 1833; Margaret 1836. James P and Catherine/Kathrine are in the 1841 and 1851 census in Strathblane, but the only children whose age matches one of the children listed in the IGI are James and Margaret, listed as aged 12 and 5 in 1841 and William and Margaret, listed as 17 and 14 in 1851, but not there in 1841, apparently. There is a William in 1841, transcribed in FreeCEN as aged 1; if this is a misreading of 7, it would fit.
So where was their daughter Elizabeth in 1841 and (even more important) in 1851? If she survived, she would have been 15 on the date of the 1841 census, and 24 on the date of the 1851 census. Is she a bit young to be the one who married James Risk in St Ninians in 1843? She isn't the one who married John Dickson in St Ninians in 1848, because that one was born in Kilsyth according to the 1851 census. There is an Elizabeth Paterson, 23, a servant in Glasgow in 1851. Could this be the daughter of Catherine Ferguson? I have not found her in 1841.
To my mind all the evidence that I can find, plus the reported death certificate that I have not seen, points to your Elizabeth Paterson being the daughter of Alexander P and Elizabeth McGowan. The only way to prove conclusively that your professional searcher got it wrong would be to find a marriage or death for the daughter of James P and Catherine Ferguson, though the difference in birthplaces should be a strong indicator. Can anyone look at the 1861 and 1871 on A******y and see where that says Elizabeth Paterson or Gillespie was born?