The Cuddy are strongly represented in a charity Ulster Historical Foundation whose website is
https://www.ancestryireland.com/ has 102 Cuddy listed in Co Antrim and down starting in 1665! yes sixteen 1665 so you have much work to do Credits required but index free. but no Laurence/Lawrence found .
i am biased to belfast so lets hope thsi works out
the earliest street directories include
http://www.lennonwylie.co.uk1806
Cuddy, John, glazier, paint, oil and glass store, 25 Church lane.
1808
Cuddy, John, glazier, paint, oil & glass store, 20 Church Lane
Cuddy, William, plumber, 13 Marlbro' Street (Marlboro')
1819
Cuddy, James, glazier & dealer in oils & colours, 25 Church Lane
is your Lawrence Cuddy b 1811 whose entry in the enumerators handwriting in the 1841 Census gives both he and Elizabeth aged 30 and although hand ticked across occupation reads with difficulty shipwright who is also on family search I realize the ages change in later census
you must be the Paul listed on
http://www.GenesReunited.co.uk/tree/results/cuddy/lawrence?memberkey=0&yearofbirthrange=0Census
1911 Co Antrim 39 for later relatives
1911 Co Down 15 for later relatives
Griffiths Valuation about 1861 only for Belfast area
http://www.failteromhat.com/post1845.php for speed
http://www.askaboutireland.ie/griffith-valuation/ for detail
Cuddy Hugh Cromac Townparks Chichester Street Shankill Antrim
Cuddy Henry Thompson's Row, Legoniel Shankill Antrim
Cuddy Elizabeth Village of Dundonald,Church Quarter Dundonald Down
1823-1837 Tithes Applotment Books in the 6 counties
http://www.irishgenealogyhub.com Cuddy, A.-T: Mobnee Y: 1827-Lissan-Derry & Lissan
Cuddy, A.-T: Mobnee Y: 1827-Lissan-Tyrone
Cuddy, Abel-T: Aughagala Y: 1827-Clonoe-Tyrone
Cuddy, Alex.-Townland : Tullintanvally Year: 1833-Annaclone-Down-
Cuddy, Alexander-Townland : Tullintanvalley Year: 1833-Annaclone-Down-
Cuddy, James-Townland: Aghagallon Year: 1834-Aghagallon-Antrim
Cuddy, James-Townland: Aghalee Year: 1834-Aghalee-Antrim
Cuddy, John-T: Killen Y: 1827-Clonoe-Tyrone
Cuddy, Michael-Townland : Tullintanvalley Year: 1833-Annaclone-Down-
Cuddy, Mrs. Townland: Antrim Town Parks Year: 1833-Antrim-Antrim
Cuddy, Thomas-T: Innish Y: 1829-Killeeshill-Tyrone
Cuddy, Thos.-Townland: Killough Year: 1834-Aghalee-Antrim
Cuddy, William-T: Mullaghmore Knox Y: 1826-Donaghmore-Tyrone
Cuddy, William-Tl: Terlugan Yr: 1827-Carnteel-Tyrone
Cuddy, Wm.-Townland: Ballykelly Year: 1833-Ballinderry -Antrim
Cuddy, Wm.-Townland: Gortraney Year: 1833-Ballinderry -Antrim
Cuddys, Henry-Townland: Antrim Gardens (Mill Row) Year: 1833-Antrim-Antrim
Cuddys, Henry-Townland: Antrim Town Parks Year: 1833-Antrim-Antrim
1823-1837 Tithes Applotment Books in the 26 counties
http://titheapplotmentbooks.nationalarchives.ie/68 Cuddy listed
end of part 1
to be continued