I think the Charles Paterson in Stirling in 1841 could well be yours. Agreed, you've been unable to find a baptism for Alexander but this Charles and his wife Jean had at least 4 daughters, possibly 5. There are no baptisms online for these girls either which is why I think Alexander could well belong to this family! I don't know if Stirling parish baptisms are patchy but I counted, for example, 11 Paterson's on 1851 census who are shown as born in Stirling 1831 to 1841 but on familysearch only 4 baptisms are shown in this period!
Charles Paterson married Janet Mustard 12/12/1818 St. Ninians, Stirlingshire. (On census records his wife appears as Jean and Jane both variants of Janet).
The youngest daughter Rachel married an Alexander Laing 13/3/1857 Glasgow and the marriage certificate states her parents were Charles Paterson and Jane Mustard.
Janet/Jean/Jane is shown to have been born Alloa, Clackmannan and there are 2 possible baptisms for her:
Janet Mustard b.27/2/1791, bp.13/2/1791 Alloa, Clackmannan dau. of Alexander Mustard and Christian Paton.
Jean Mustard b.15/11/1792, bp.18/11/1792 Alloa, ditto
As to Charles - his age on 1841 census suggests birth date 1792-1796. There are 2 males baptised at St. Ninians to the same couple but Christian names are not shown:
Paterson (male) bp.4/9/1791 St. Ninians, son of Alexander Paterson and Helen Steven
Paterson (male) bp.16/8/1793 ditto ditto
Since Charles married St. Ninians, one of these 2 baptisms is probably him. Following the Scottish tradition of eldest son named after paternal grandfather, this could explain how your Alexander got his name.
I know this is all if's and maybe's but seems a strong scenario to me when there are no baptisms for the known daughters of Charles either.
Annette