You are correct it looks like fungi growing on the dead parts of the moss.
But then again, as @kevinsky pointed out
This could be the result of incorrect watering leading to salts being deposited on to the moss.
Moss needs high humidity and it would die back when facing extreme temperatures with in a few hours.
You should cut/remove all the dead plant matter to avoid further rotting.
Then simply mix a table spoon of bicarbonate soda with 1 liter of clean water and spray the whole of the affected area.
For a stronger solution you could add a tiny drop of neem oil but warning it must be diluted as it can harm your plant and it smells so bad
The thing most people dont realise about Bonsai trees is that every year or so, the soil should be removed so that the roots may be spread out to be inspected and maintained, by removing the dead roots and trimming them to fit in the pot, without looping around the pot.
This also may be the time for some roots to be shaped/exposed further to create the overall look of a miniature giant tree.
The soil or growing medium would also be replaced during this time.
And you guessed it,
Most of the moss would also be replaced or trimmed back depending on what is necessary.
To take care of a Bonsai you will need some basic tools made for the job.
Also the plant pot may need changing only if you wish for it to grow a bit bigger.
To debunk the rot theory
You will need a cheap and cheerful humidity meter that registers the maximum and minimum levels to see exactly what humidity extreme its struggling with if any.
And get a soil humidity meter, cheap as chips will do as long as it's a real active sensor that actually works . UPDATE: get a German soil humidity meter
Although its unnecessary it might help reduce he possibility of over watering when left with caretakers.
In london I only need to water my pots about once a week or so but my plants are still tiny seedlings.
Dont forget the plant may still be tied down with wire by the roots so ensure you check that before yanking it out.
Dont yank it out :)
No worries :)
Lots of vids online
In fact you have a famous Australian Bonsai fanatic on the tube, probably because there are so many native Australian Bonsai it's so easy to get hooked :)