The area was not anywhere near as built up as it is now. Hamilton Lane would have either come under the parish of Littleham or the parish of Withycombe Raleigh.
If you have access to the census of 1841 - 1851, a trawl back and forth should show it up (and the people who lived there). A word of warning though the address might not state 'Hamiltion Lane' as such!
If you don't have access to the census let me know and I'll have a go.
There is a small local museum in Exmouth - if you look them up they might be able to give you some background. I've found them very friendly and helpful. There were, and are a lot of pubs in the area, and they were called different things, Inns, Taverns, Victuallers, etc (just to confuse the issue!).