Green’s Dictionary of Slang

dog n.6

1. (Aus.) food.

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[Aus]Baker Popular Dict. Aus. Sl.

2. (Aus.) a drinking debt.

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[Aus]Baker Popular Dict. Aus. Sl.

3. (US) a state of drunkenness.

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[US]F.J. Wilstach Sl. Dict. Stage 26: An intoxicated person has a ‘bun’ or a ‘dog.’.

In phrases

kill one’s dog (v.)

1. to be drunk, to drink heavily.

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[US]B. Franklin ‘Drinkers Dictionary’ in Pennsylvania Gazette 6 Jan. in AS XII:2 90: They come to be well understood to signify plainly that A MAN IS DRUNK. [...] Kill’d his Dog.

2. (Aus.) as let’s kill a dog, an invitation to drink.

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[Aus]Aussie (France) 9 Dec. 19/2: The big Melbourne Show is in sight, and it’s going to be a ‘Dry’ Show. That means that the visitors can’t ask each other, ‘Can you stop one?’ or ‘Can you keep one down?’ or ‘What about it?’ or ‘Let’s kill a dog.’.
on the dog list [ety. unknown]

(Aus.) barred from drinking.

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[Aus]Sun. Times (Perth) 28 Feb. 3/1: His first crushing was rich enough to buy him sufficient beer to get him put on the prohibited list. His second parcel showed improved values, and he got drunk in defiance of the ‘Dog List’.
[Aus]Dly News (Perth) 22 Mar. 7/3: ‘The policeman asked my wife if she wanted me put on the “Dog” list,’ McDonald said. ‘She said: “I can't — because he doesn’t drink beer”.’.
[Aus]Dly News (Perth) 1 July 13/3: One desperately miserable woman found that ‘putting a man on the dog list’ was a ridiculous farce.