Plenty of bench meters go down to the uA level (random example: Keithley 175A's lowest range is 200uA with 4.5 digits, so 199.99uA full-scale, or Keithley 199: lowest range is 30mA with 5.5 digits, so 30.0000mA full-scale). Haven't checked the specs for the accuracy, but I would expect it to be accurate to at least 1uA. These are sometimes available used for under $100 US, but something like the Fluke 87 used will also cost much less than $700 (AUS?). Of course bench meters often lack features like continuity, capacitance and even diode test, so they might not be great as only DMM. And buying used does carry some risk and means you don't have warranty. Bench meters often have less overload protection, since they don't tend to be used in industrial environments; few of them have HRC fuses.
You can probably get 4.5 digit meters with a 1uA resolution from lesser brands like Extech or Amprobe for under $250 AUS new. Keep in mind that 1uA precision (i.e. it shows enough digits to resolve 1uA) doesn't mean it's actually accurate down to 1uA. For example, the Fluke 87-V has an accuracy specification of 0.2% + 4 digits for the 600.0 uA range. So for a 1.0uA reading, the tolerance is +/- 0.4uA, i.e. between 0.6uA and 1.4uA. Cheaper meters are likely to be worse.