Hi Russell from a fellow Victorian.
I'd agree that it is worth hanging out for at least 300mm for a birding lens. I used my Tamron 18-250 for years for birds but was often frustrated by the lack of reach. A really really sharp 200 might be good enough to allow heavy cropping (or with a good teleconverter if you have one), but you won't get that for $100.
Much of your low budget will be swallowed by shipping costs if you buy from overseas. I'd suggest looking on ebay au or gumtree (if there isn't one from an Australian seller in the PF marketplace).
Some people use MF lenses with catch-in-focus for birding, but I find AF is really useful for birds. You should be able to pick up a Sigma 70-300 or one of the other budget AF options people have mentioned for $100 or less. Here are some examples from Gumtree at present:
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Sigma Lens 70 - 300mm DG Zoom and macro to suit Pentax Camera | Lenses | Gumtree Australia Salisbury Area - Burton | 1094306763 - Sigma 70-300 asking $50
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sigma zoom lens 100-300. 28-90. for k mount pentax $80 each | Lenses | Gumtree Australia Ryde Area - Ryde | 1092339291 - Sigma 100-300 asking $80
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Pentax camera | Non Digital Cameras | Gumtree Australia Brisbane North West - Ferny Hills | 1098224549 - another Sigma 100-300 asking $50
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Sigma Camera Lens AF 70-300/4-5.6 MS2 F/Pentax | Lenses | Gumtree Australia Tablelands - Atherton | 1096682835 - Sigma 70-300 asking $100
On Gumtree there are a couple more Sigma 70-300 lenses for sale in Melbourne asking $120. They might do them for less and you could possibly pick up.
See this comparison:
https://www.pentaxforums.com/forums/10-pentax-slr-lens-discussion/57560-teste...-70-300-a.html (The reviews of the Tamron equivalent all refer to its purple fringing.)
I agree that the 55-300 is the pick of the consumer tele zooms but these lenses (even in DA-L form) seem to go for $200+ in oz at present. The Sigma 70-300 would be great value at less than half the price.