I am researching COULTER from Dundonald/ Irvine Ayrshire.
Unfortunately i am not back as far as 1803 yet.

However, i also had an illegitimate ancestor who carried her fathers surname not her mothers.

We were eventually able to confirm who her Father was through the Kirk mins.

The Kirk/ Church meeting minutes are held at NAS (National Archives of Scotland) in Edinburgh; if they survived.
Single mothers were called before the Church Elders and questioned/ interrogated as to who the father of their child(ren) was, the reason being that the church was unwilling to financially support these mothers if they could make the fathers do so.

The following link is a search engine you can use to see whats available for Dundonald.

http://www.dswebhosting.info/NAS/dserve.exe?dsqApp=Site20&am...

Good Luck
Debbie