Dear Amanda
ps I did a place search of LDS family search:
Topic England, Cumberland, Cockermouth - Church records
Titles Births and baptisms Independent Church (Cockermouth)
Bishop's transcripts for All Saints Church, Cockermouth, 1689-1862 Church of England. All Saints Church (Cockermouth, Cumberland)
Church records, 1841-1910 Wesleyan Methodist Chapel. Cockermouth Circuit (Cumberland)
Cockermouth miscellanea Cumbria Family History Society
Parish registers, 1632-1881 Church of England. All Saints Church (Cockermouth, Cumberland)
Parish registers, 1865-1883 Church of England. Christ Church (Cockermouth, Cumberland)
You may be able to search these records at the LDS centre closer to you, I see they have no RC records.
They also have all the censuses 1841-1891 for Cockermouth
Pat