If the death of Alexr in 1784 is a child then the Alexander/Mary gray child born 1782 could be this one. Is there any way I can verify that? Otherwise I'm a bit stumped as the only other Alexanders are born in Cambusnethan 1783 and Cadder 1790. I could check on Church of the Latter Day.
Unlikely. And there is no point in looking on the LDS web site for BMDs if there is nothing more in Scotland's People because what the LDS (and the various commercial web sites) have is indexes to original documents which are only available on Scotland's People.
The only other possible source would be a gravestone, if one exists. There are published booklets of pre-1855 monumental inscriptions in Lanarkshire.
If I were you I would get access to the film of the Carluke parish register and go through it, extracting all the Lightbody baptisms and marriages (use the index at Scotland's People to make sure you get them all) and see if they list residences and witnesses. Look also for Lightbodys cropping up as witnesses to other people's baptisms, and make a note of their residences, and see if you can build up a picture of where in the parish they lived.
To get access, assuming that you don't live anywhere handy for Lanarkshire and you don't want to pay 6 credits each to look at the 65 baptisms and 31 marriages of Lighbodys there, you need to go to any LDS Church Family History Library near you and arrange for them to rent the films of the Carluke parish register. There is a small fee for this. When the film arrives, take a day or two to go to the library, sit down and go through it carefully.