Even if you use the display only when, say, a switch (pushbutton) is closed, LED's will be too hungry for a watch battery.

You can buy a 4.5-volt cell at most drugstores here in the U.S. so I imagine it's the same in Oz. If you (a) use a clock display LCD, and a 3-volt microcontroller, of which there are quite a few, and a CMOS LCD driver, perhaps even one that is only active when you switch the display ON, you can probably do this with increasingly common AA-sized lithium cells commonly used in digital cameras, etc. With proper circuit layout, and proper design, you'll have little trouble with the power supply, as you won't even need a voltage regulator, and you'll need relatively few components if you find the right ones. I'd recommend you start by looking for an LCD.

DON'T USE A CHARACTER DISPLAY ... if you want to be thrifty with power, you need to use the power-thriftiest device you can find.

The easiest ones to use are quite large, as they have nominally 1-inch tall digits, but they also are convenient as they are Dual-In-Line packaged, typically with 40 pins, so you can use 20-pin SIP sockets for them. If you're clever, you'll find a multiple backplane (multiplexed) display. You have to search, but they are available. You drive the LCD by setting a 7-segment value representing the digit to be displayed to the appropriate pins, and then toggle the backplane on while the digits are off, and then the backplane off while the digits are on. With four digits and four backplanes, you use only 11 pins to effect this interface. Of course, you'll need one input to tell the MCU that you want to see the value to be displayed.

RE