Can anyone recommend me a place to get a 60a 5 speed rebuilt in Melbourne?

depends if you want done cheap or good.

pm revhead- he might know some places other than work for a stockish rebuild.

What is wrong with it and what is your budget?

It catches on down-shifting into third and second, unless driving quite slowly.
Third is worse than second. By rev matching is usually slips down into second ok. I just avoid shifting down to third.

I'm prepared to pay what it takes to fix it properly (within reason).

First things first. What weight oil did you use in the gearbox?

I used 75-85W manual transmission oil. Castrol VMX-M I think it was called.

It is definately the right oil. Judging by your description it is most likely going to be selector hub or synchro ring related. If you bring the box to me I can have a look and it might be a simple fix, or it could need a couple of synchros. Unfortunately, for a full rebuild I don't do them quickly, but a quick repair in the shed is doable. Feel free to give me a buzz if you want.