We asked the ATO if tax payers could just claim wrongly charged GST back on their BAS, unfortunately it's not that simple:
Thank you for your email.
If the registered supplier has incorrectly treated an entity as an Australian consumer and GST was charged as a result. There are two options to get the overpaid GST back.
Division 142 of the GST Act sets out the rules relating to refunding excess GST. You can ask the supplier to reimburse you for any overpaid GST, they can then make a decreasing adjustment when they lodge their BAS and claim the overpaid GST back.
You can refer to GSTR 2015/1 for more information, in particular, paragraph 18 explains a decreasing adjustment when a supply stops being taxable.
If no reimbursement is made and you are registered for GST, hold a valid tax invoice and meet all of the other requirements of the GST law, including the special rules in Division 142 (see GSTR 2015/1), then you are entitled to the input tax credits in relation to the acquisition of the thing supplied if the acquisition was a creditable acquisition. This means that you can claim the input tax credits by making an increasing adjustment when you lodge the BAS. Please visit the following link for more information on claiming input tax credits.
Although a non-resident is not required to issue a tax invoice to their customers, they can still issue tax invoices if they want to, provided they have an ABN and are registered for GST.
If a sale is below $82.50 (including GST), the recipient is not required to have a valid tax invoice in order to claim the input tax credits for the purchase.
Kind regards[Dennis from the ATO ...]