I agree with you and the Magistrate that the only way to sort this mess out is to appeal it to the District Court.
Notwithstanding that you have already paid $8K off the $13K judgment (leaving $5K or thereabouts still to pay) , you never should have had judgment entered against you for the $13K because you say that the terms of the transport contract were never varied.
Because the amount in dispute is more than $10K, this is not a minor case and you therefore have a right of appeal to the District Court.
You should plead that the entire judgment of $13K should be set aside, and that the $8K that you paid following that judgment should be returned to you.
You have 21 days from the delivery of the Magistrates Court judgment to appeal ; leave to extend this time can be sought if needed.
See DISTRICT COURT RULES 2005 - REG 51 Appeal, commencement of http://www.austlii.edu.au/au/legis/wa/consol_reg//dcr2005236/s51.html
This will also be helpful to show how an appeal is set out. It will have to be changed to suit the specifics of your case.http://www.districtcourt.wa.gov.au/_files/Circular%20to%20Practitioners%202009-2%20%20Registrar%20Appeals.pdf
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