How about some of those alternate spellings?
Only a Dunice family 1841 UK Census, Nottingham identified as Irish born.
My computer isn't connecting to the Donegal phone book - on line. The BT/NI telephone book shows no spellings of
Dunniece, Denice or Denniss probably a different
spelling and phone books are spelling specific so you could try again with different spellings.
The Donegal phone book would give you the surrounding counties as well if you get the spelling right.
Barb Denice 10, living Shankill, Antrim Church of Ireland daughter to Mary Meneicie born Belfast
Rose Denice - age 5, RC born Roscommon -looks like an orphanage on Merrion St. Dublin
Jane Denneice civil registration of birth July-Sept 1888, Londonderry #Vol.2 Pg. 180 #101062/004193976