most sites will use a normalised name in the index (otherwise how do you handle cath, kate, cathArine, cathErine etc) so don't worry on this point.
Have you looked for an obit in the Warrington Guardian ? - Warrington I know has an excellent local history section in the public library which should be online.
Did they marry in Ireland ?
The Manx in Liverpool kept close links with the Island so it is possible the family were confused over the place of her birth