Some likely shipping companies in the cattle trade were:
Belfast, Mersey & Manchester S.S.Co.
British & Irish Shipping Co.
but my lists don't say which were specifically cattle boats. They would have been designed or modified for livestock transport probably on one deck, from which the animals would be driven 'on the hoof' down a gangway to the landing stage.
The full fascinating story of 100 years of Birkenhead's and Wallasey's involvement in this important trade is told in the book 'Meat at Woodside' by Ken McCarron (NMGM/Countyvise, 1991). At its height, late 19th, early 20th century, over half a million sheep and cattle were imported. It was killed off by the refrigerated carcase trade from worldwide.
Another part of Birkenhead's past, gone for ever!