In Australian English, Bugs Bunny ¹ is a derogatory nickname for the metal eagle at the top of the Australian-American Memorial ² in Canberra, Australian Capital Territory, and by extension a derogatory nickname for the Australian-American Memorial.

¹ Bugs Bunny is the name of a U.S. animated-cartoon rabbit conceived in the late 1930s at Leon Schlesinger’s animation unit at Warner Brothers studios.

² Erected to commemorate the assistance of the USA in the Pacific theatre during the Second World War, this memorial is a column surmounted by an eagle with wings upswept in a victory sign, and the nickname Bugs Bunny alludes to the fact that, from a distance, the eagle’s wings look like a rabbit’s ears—as explained by Stephen Mills in We were never the same again, published in The Sydney Morning Herald (Sydney, New South Wales) of Tuesday 15^th August 1995:

Nine years after the war’s end, grateful Australians built a 73-metre-high steel and aluminium tower in Canberra, capped it with a three-tonne eagle, and dedicated it to the Americans who had helped defend Australia.
Who was chosen to perform the opening of the Australian-American War Memorial? It was 1954, and without any apparent sense of irony or contradiction, we asked the Queen. If ever a symbol was needed of our ambivalent, confused relationship with the US, then surely that was it.
[…]
[…] The uplifted eagles [sic] wings make a V shape, to celebrate victory, but the whole design from afar looks like an elongated rabbit, giving rise to the nickname “Bugs Bunny”—another jibe at the American culture we love so much and cannot live without.

This photograph of the eagle was taken in 1953, during the construction of the Australian-American Memorial in Canberra, Australian Capital Territory—source: Trove – National Library of Australia:

 

The earliest occurrences of the nickname Bugs Bunny that I have found are as follows, in chronological order:

1-: From the account, by Ian Healy, of the unveiling of the Australian-American Memorial on Tuesday 16^th February 1954, published in several Australian newspapers on Wednesday 17^th February 1954—for example in The Daily Advertiser (Wagga Wagga, New South Wales):

More than 4000 people attended the unveiling of the memorial, the first ceremony for the Queen after the Executive Council meeting.
The controversial Memorial, a 258ft. sandblasted aluminium column with an eagle on top, glittered in sunshine on the slopes of Mount Russell.
Americans in Canberra say the Memorial is “swell” and “a mighty monument”, but Australians are not all so enthusiastic.
The eagle has wings upswept in a V-for-Victory sign and some locals, who claim the wings look like a rabbit’s ears from a distance, have dubbed it “Bugs Bunny” and “Brer Rabbit”.

2-: From The Canberra Times (Canberra, Australian Capital Territory) of Saturday 19^th September 1964:

Canberra in a nutshell

Brevity, it is said, is the soul of wit.
It is also quite a serious business for Canberra’s cab drivers.
Most of their trade is picked up by radio calls, and the driver with the quickest finger on the radio button and the briefest description of his location usually gets the job.
This is a short glossary of cabbies’ terms for Canberra locations:
Gasworks—Parliament House.
Bugs Bunny or The Pelican—The American War Memorial with its metal eagle. Used to identify a call to the Russell Offices.
Pentagon—Alternative name for Russell Offices.
Hotcan—Hotel Canberra. Other hotels have similar names; Hotwell (Hotel Wellington), Hotrex (Canberra Rex), and so on.
Hillside-Rex—Hillside Hostel. Sometimes known as Hillside-Hilton.
Eskimos—Academy of Science.
America—American Embassy. All the embassies are designated by their countries. Russia, Germany, France, etc.