I think some points have been lost or misrepresented...
A subsidy itself will not lower the price. It merely allows domestic producers to compete with foreign producers at the market equilibrium price. The world price is not influenced by a subsidy in Australia. So to say a subsidy will reduce prices is false.
For example, without a subsidy the world price for wheat is $10 a tonne. At this price Australian producers are willing to supply 50% of our needs, and we will import the rest. With a subsidy, the world price would still be $10 but now more Australian producers would be willing to enter the market and they would supply say 90% of the market.
So all that a subsidy does, is allow more domestic producers to enter the market, or thought of a different way, it will allow domestic industries to remain open when foreign competition increases (which may drive the price down). For example Australia gives generous car subsidies because if they didn't australian industries would close as they would not be able to compete with the cheaper imports. But you notice that cars are not cheaper in Australia as a result.
I think where the confusion is coming from is that tariffs ARE inflationary because they increase the price. So a subsidy is a form of protection that is not inflationary.