I'm an Australian citizen. I have two government issued photo IDs, a Proof of Age card issued by the Australian Capital Territory Road Transport Authority (they issue drivers licenses, you get a proof of age card if you are 18 or older and don’t have a drivers license) and a passport issued by the Australian Government Department of Foreign Affairs and Trade.

My proof of age card is my primary ID. My passport is kept secure and only comes out for two reasons, checking in for transport to an international destination and immigration procedures when entering/exiting a country. As an airport security checkpoint is neither of these and the TSA has no immigration powers whatsoever, they have no right to see my passport.
However, the TSA claims that my proof of age card, the only ID I have that they have even a remote right to look at, is not valid.

Is there any type of US issued ID that I might be able to get (without needing to luck out for a green card in the lotto or move to Canada for three years before getting a NEXUS card)?

It has to be US government (whether state or Federal) issued ID or a passport.

So learn to drive & get a US driving licence (sorry license) or refuse to produce ID & get felt up by someone each time

Just use your Passport.

Although TSA has no immigration authority, they also don't bother training the travel document checkers on anything besides US/Canada drivers licenses and passports for all other countries.

Really the ID check should go away and this would no longer be a problem.

That's the thing, I don't think the TSA thinks it's a problem. They likely feel that giving the business to people with Passports from countries with predominately brown people is a brave, bold and patriotic move.

Right up there with confiscating 3.5 oz toothpaste.

You don't need to learn to drive. In most (so far as I know, all) US states, if you can prove residence the state Department of Motor Vehicles will issue you an ID card that looks like a drivers license, usually with the notation that it isn't a drivers license. Most people trained to check IDs know about these and will accept them.