I took ApkInstaller from my internal memory and zipped it up and attached it for you. Unzip the file and move to your SD card, touch it and choose Install. It should install on your device I the Fire is an Android device.
Do you have Astro File Manager? I can get that for you as well. Or what ever file manager you want from the market.
This is an interesting problem, you are in Australia can purchase the Fire, but can not purchase from Amazon or the Market.
You can access the Market which means the apps listed have links to the developers. Once I had to go to the developer link in the Market and get the app because the developer was in England. They verified I had paid the Market and then sent me a link to download the app. I do not remember why I could not download the app, Titanium Backup licence, but it worked out for me going direct to the developer.
Can you get a Paypal account? Most developers will sell directly to U, if you donate via Paypal.
Let me know if the apkinstaller works for you. I may have to get the file using the ADB commands, if this copy, past, transfer did not work.
I was going to attach Astro, but I have several counting plug in's and licences, so I have to find the main file on my phone tomorrow. It is past my bed time here.