I've never heard this use of cod either so I looked it up. In OED it gets noun definition 5 of 7 with only one related earlier use. It was in a 1952 theater dictionary so maybe there's a connection there:
    1. A slang appellation applied to persons, with various forces: see the quotations.

c1690 B. E. Dict. Cant. Crew, Cod, also a Fool.."An honest Cod, a trusty Friend." 1708 MOTTEUX Rabelais V. v. (1737) 18 O what an honest Cod was this same Ædituus. 1851 C. D. BEVAN Let. in Beddoes' Poems & Lett. (Introd.) 130 [At the Charterhouse]..In those days the pensioners (or as we called them 'Cods') were not remarkable..for cleanliness. 1855 THACKERAY Newcomes II. 333 The old reverend black-gowns..the Cistercian lads called these old gentlemen CoddsI know not wherefore. 1873 Slang Dict., Cod, to hoax, to take a 'rise' out of one. Used as a noun, a fool. 1878 MACLEOD Hist. Dumbarton II. 46 Ye vile drunken cod.

    2. A joke; a hoax, leg-pull; a parody, a 'take-off'. (See also E.D.D. n.5) Also attrib. or quasi-adj., parodying, burlesque; 'mock'.

1905 Sketch LI. 472/2 Says he: 'Is that an absolute bargainno cod?' Says she: 'I don't know what the fish has to do with it, but I am perfectly sincere.' 1914 JOYCE Portr. Artist (1916) i. 45 Some fellows had drawn it there for a cod. 1952 GRANVILLE Dict. Theatr. Terms 46 'Cod version, a burlesque of a well-known play.' 1959 Church Times 16 Jan. 4/4 'The 'cod' Victorian decorations tend to disguise the editor's underlying seriousness.' 1959 Listener 29 Jan. 228/1 She obliged, initially in the delicious hiccup polka, a cod of Old Vienna. Ibid. 228/2 Joyce Grenfell too, doing her evergreen cod chorister. 1961 B. WELLS Day Earth caught Fire ii. 31 Pete picked up the empty tea mug and again used it as a cod mike. 'Alcoholics of the press, unite! 1962 Listener 5 July 36/1 The very idiosyncratic cod cockney of the scenes. 1970 Guardian 11 May 8/2 The cod version of 'Road to Mandalay'.