Well, I know I won't need snowshoes for much of the trip and would rent them once I get there, if you mean the sport snowshoe you strap on over your regular shoes. (Our English language is kind of funny when you think about it - snowshoes are sometimes very different than snow shoes, shoes that will protect our feet from snowy weather.)
If you mean thick, warm shoes that can take cold weather I would probably wait and buy when I arrive. You would find more models than in Australia, maybe cheaper.
If you already have them, wear them. They weigh your luggage, not you and your clothes

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I'm going to have to study your itinerary some more because it looks like you will be checking in for flights on several different airlines at different times. That means different airlines' policies and fees will apply.
Maybe some of our knowledgeable members here can help. I understand this is a somewhat complicated trip and that you are not a frequent traveller - we all started out that way.
I think some of of our members think you are playing with us and do not understand you are really and honestly asking for our help here because they are confused with how you think and write your thoughts.
Those who want to help will, and we have some frequent and knowledgeable members who can be helpful. Just let me ask you to be patient with us so we can understand, and we must also be patient.
Those who can't will just not post, because not all of us have the patience. That's OK and we have some time before your trip.
Meanwhile think of ways you can make your trip fit into one suitcase or bag, weighing 23 kilos or less. It can really be done, even in winter. One really important thing I've learned is how to be very critical about my packing - it has saved me lots of money and trouble over the years. Trying to get around with my carry-on stuff and TWO pieces of luggage is my travel nightmare.
You can wear your heaviest shoes, carry / wear your jacket (wear it there, carry it in Australia and Singapore), etc.