An e-mail enquiry to Whitehaven Records Office should be fairly cheap. At worst, they could say "no". http://www.cumbria.gov.uk/archives/recordoffices/whrec.asp
Geoff,
Personally, I would suggest that regardless of the religious denomination, for post-1855 burials that took place in Whitehaven, in most cases the easiest and best way to find the information is from the Whitehaven Cemetery records. As you have intimated to fellow researchers these are accessible at the Whitehaven Records Office, Scotch Street, Whitehaven. In addition, if there is a surviving memorial / headstone almost all of these have been photographed by a volunteer and saved digitally.
For those unable to make a personal visit or ask for a voluntary 'look up' request there is, of course, a fee payable for the staff to do a search (£30). There would usually be a fee, or at least a request for a donation to ask just about any archives, museum, parish church etc to look something up. If someone wants an actual birth, marriage or death certificate then ancestry researchers are required to pay for these. One might then quite reasonably expect to make a donation to the parish office of a church to look up information directly from their church registers.
One thing to bear in mind when trying to find the location of Catholic relatives in the Whitehaven area, particularly after 1880, depending on where they lived they may not be buried in Whitehaven Cemetery. For example, if they lived in the Hensingham district they may have been buried in Hensingham Churchyard or the 'new' Hensingham Cemetery.
If a Catholic family lived in the Moresby / Parton area they may have been buried in Morseby Churchyard (St Bridget's). There were some Catholic burials at Moresby Churchyard before 1880. After that date, as undoubtedly many researchers will already be aware, the Burial Laws Amendment Act allowed burials in a churchyard without having to have received the rites of the Church of England. In the case of
celticmoon's ancestors (living on Catherine Street 1870 - 1890) my feeling is the most likely place for interment would have been Whitehaven Cemetery.
Incidentally, not all the West Cumbrian Catholic parish records have been transcribed and / or donated to the Whitehaven Records Office (e.g. St Joseph's, Frizington). Likewise, there are some Church of England burial registers that have not been transcribed and / or donated to the Records Office (e.g. the Lamplugh parish burial registers after 1887).
Hence, if one wished to look for information in one of these registers it would likely entail either making an appointment to visit the parish in person or request someone from the parish to do it. When I was looking for information from these registers that is what I had to do. Not everything is available online.
Attached photographs: St Begh's R.C. Church, Whitehaven.