Lawrence not a typical Irish name......I have not heard that before.
Not clear to me what you mean here by Laurence not being a typical Irish name.
Maybe not as a surname, but quite Irish as a Christian name - for St. Lawrence O'Toole, archbishop of Dublin. One of very few (two?) canonized Irish saints before the modern era. The many hundreds of others being saints by acclamation rather than canonization. Patron saint of Dublin.
Given that the family surname is Toole, Lawrence seems very apt.
https://www.newadvent.org/cathen/09091b.htm
Wexflyer that is an interesting point about St Lawrence O'Toole archbishop pf Dublin. My tattered book of Saints says he was born at Castledermot in Co Kildare. This is about 47miles from Dublin. St Lawrence O'Toole was born in 1128 and so the name Lawrence has been around since then!