As to the question of common caliburs, I’m fairly certain the market would choose a mix, just as it does in other areas. Take computing, for example. Most personal computers operate under the Windows on an Intel chip. Some people buy MacIntosh, some people install Linux, and this margin of early adopters and tinkerers keep that central standard on its toes without fragmenting the platform beyond a level of essential utility. I suspect the same would be true with firearms: economies of scale being what they are, most people would use the cheaper, easier to use, more popular guns and ammo. But you’d still have a margin of tinkerers and early adopters looking for something more effective or just plain cool, testing out other ideas.
As to where average citizens would receive military training in more advanced weaponry, my thinking is that the Anarcho-Capitalist assumption is that people aren’t complete morons. Even in the absence of a coercive state, you’d still have a lot of people seeking to support some kind of military establishment. A community’s top level military capacity could be a voluntarily funded corps that would act as a “first response” force, while the people at large are busy getting themselves ready to join the fight.
To address your own scenario, we have, first, to address the likelyhood of the scenario, and the precise form it would take if the US were an anarchist zone rather than a semi-republican police state. You have to ask the question, why are they invading? There’s no need to knock out our ability to respond if they should attempt to assert control over Taiwan, for example. There’s no US president backed by a US military to selectively press for human rights reforms. There’s no immigration policy preventing the peaceful arrival of Chinese immigrants, nor a trade policy preventing Chinese merchants from trading peacefully or Chinese businessmen from establishing operations. There is very little incentive to invade in force.
And even if they did, there’s no central authority to knock out with military force; any invasion faces the question of “where do we strike?” They could temporarily take territory, but the expense of maintaining that beach-head (this is a trans-Pacific operation, after all) would most likely be more than could be extracted from the area. In the absence of psychologically satisfying victories against clear targets, I wonder how long the Chinese government could justify the expense, before the drain on resources emboldened their own foreign and domestic enemies. It was Chinese military theorist Sun Tzu who warned that protracted warfare weakened the State and emboldened what is sometimes translated “The Feudal Lords”. It was the Spartans who discovered the hard way that when the soldiers are away, the Helots revolt.
Finally, a free society is heavily enmeshed in trade with its neighbors. The Chinese would face not only higher expenses, but declining revenues as a result of disrupting trade between the former United States and China.