I went to Kosciuszko national park with my pug, he was not allowed and we had to leave him at a vet with kennel facilities, if you get caught it is a few hundred $$$ fine, and I believe they said 3 or 4
Do some google searching for places in the snow or public dog paths that you can drive up to, then give them a call... providing they don't have summer paths that they close, you may find something where you can park and play with the dog.
I cannot stop you, if you want to hide the dog in the car and risk a fine and take him/her to Kosciuszko national park... but I do not recommend that. I did notice many stops going up to Perisher to snowboard where we could have let our little boy play and experience snow, but we did not risk the fine...
Also there is no shade at perisher, so I advise you go in the morning early (or put up a shade) as it may be cold out, but my van was cooking at mid day (this time last year) when I went up, although we did have a very sunny day that time. May not be hot enough to make him ill, but might have been uncomfortable for a pug to be locked in a van with a higher temp than what he is used to for hours on end.
Do not forget as there is no dogs, and many rangers are about, getting him/her out for the loo will also be noticeable if you decide to risk it and you will need to check in on them, check the alarm is not constantly going off etc... again I am not condoning this, but I am more worried about safety if an owner wanted to risk it, and read this forum...
I am not assuming you do not know all this, just in case someone reads and is a new owner or has not been to the snow etc
hope this all helps :-)
PS anyone reading this, if you need info on the vet who can take your beautiful boy or girl in, please post and I will find the number, also I had a number for a sitter which took us an hour or so to get through asking and facebook... only problem was she is fully booked during the snow season