Thank you to everyone for all their help. There seems to be a lot of info on this person.
Am I sure he was born in Scotland? No. He said he was in the 1851 census, which is the only one I can find, as he died in 1858. His wife also said she was born in Scotland, and went on saying that in subsequent censuses.
On his joining the Essex Constabulary, I can find two documents, one saying he was born in Scotland and the other Longford, Ireland. (I rule out that there were two Robert Beatties in the Essex Constabulary from the army, born in 1809 with the right number of children). I notice on one of the army papers that he said he was from Scot's Corner, Clonguish, Longford -- could that have meant that there was some kind of enclave in Ireland that regarded itself as Scottish? Or could he have meant that his "ancestry" was Scottish, though he himself was born in Ireland (except that that would have to have been true for his wife as well). Or could he have meant that he was in a Scottish regiment?
Anyhow, it certainly does seem that this is the right person. I followed the activities of the 91st as far as possible on the Internet, and it seems that they were travelling a good bit around then -- St Helena, Jamaica, India -- and as was noted above, he seems to have been just about the ohnly member left on the Isle of Wight. Might he thus have had some kind of staff job? He was a staff sergeant.
Thanks again for all your help.