Another source of information on Captain John Westmacott is the Henry Moore Institute Archive in Leeds, which has quite a lot of Westmacott papers and works of art. They have some letters Captain John wrote while serving in Spain and after his return from there. Dreadful handwriting!
Their catalogue also gives his birth as 1783. It says he died from wounds received at the assault of Halifax, a confusion, I think, between his wounding at the siege of Badajoz, Spain, in 1811, with his fatal wounding by two thieves he apprehended in Halifax, Nova Scotia, in 1816.
The birth year of 1787 comes from him being said to be 29 on his memorial in Halifax. I've never found anything to corroborate this and his brother Frederick was born in 1787 so I've assumed it's an error. Always open to correction if more information comes up, of course!
Drosybont