Eric Partridge gives the following in his A Dictionary of Slang and Unconvential English (Supplement):
pig's ear occurs in D. W. Barrett, Navvies, 1880.—2A 'side light colour signal' (Railway, 3rd): railwaymen's: C. 20.—3. A blunder: mostly middle-class: since ca. 1945. Elizabeth Hargreaves, A Handful of Silver, 1954l. '"I've made a real pig's ear of it, haven't I?" said Basil, with an attempt at lightness.' (Laurie Atkinson, who asks, 'Rhyming s. for smear?')
In the main edition, "pig's ear" is given as rhyming slang for beer.
Richard A. Spears, in his book Slang and Euphemism gives:
Pig's ear! 1. an exclamation, a euphemism for *Pig's ass! (q.v.) 2. beer; a glass of beer. Rhyming slang.
And since he mentioned the source of the euphemism:
Pig's ass (also Pig's arse!) an exclamation. For synonyms see 'Zounds! [widespread slang, 1900s]
which caused me to look at Partridge's definition of "Pig's arse":
Pig's arse A low c.p. of dissent or disbelief: Australian: since ca. 1945. An adaption of pig's eye — which, by the way, had itself > Australian by 1945.
and then to
pig's eye, 2, was, by 1959, dead. (Leechman.) —3. To convey an emphatic negative, thus: 'In a pig's eye, you could!': Canadian: adopted, ca. 1945, ex U.S. (Leechman.) The phrase (in a pig's eye* was orig. euphemistic for in a pig's arse or ... arse-hole, as in a bawdy song current long before 1940. (Am. correspondent.)
Given all this, it is tempting to conclude that a "pig's ass" would represent something pretty foul.