I read that in order to register an Australian domain name, you need to have a registered business in Australia. Does anybody know a way to bypass this?
Some companies may offer a Trustee Service where they will use their information for you. Consider trying Hexonet as they usually do that. But, giles is correct that an ABN is required to register .com.au domains.
Yes, you can bypass the business address section. You can buy such domains from any reseller of ResellerClub.com and they may help you.
To own a .com.au or .net.au you must be an Australian registered company, or business with a registered business number (ABN, BRN, BN).