First, they wouldn't have much of a motivation to do so after a cost/benefit analysis. Second, their subsidy is not unlimited. You can't squeeze blood from a stone; eventually the people cannot be taxed or the currency inflated any further, and the military goes broke.
And their free-market opponents do not have a "private military". They have a populace armed to the teeth, inexpensive and easily obtainable defensive hardware, and who knows how many dozens or hundreds of decentralized insured and reinsured defence agencies, none of which can surrender in the name of the others.
Additionally, the state is fighting for an inherently divisive political goal, for which they must garner a substantial number of supporters in their own country. Once that support is lost, they face the possibility of rebellion at home, which they must fight with an increasingly weakening economy.
Also, why are we assuming a stateless country being attacked by states? It is far more likely that the stateless society will form within and among the statists via the countereconomy. In such a case, matters are made even more difficult for the state attackers because their enemies are mixed in amongst their supporters. And as the state's money becomes weaker and the countereconomy becomes stronger, they face the inevitability of their own employees defecting to the anti-statists. If a soldier can't buy a loaf of bread for his family with the government's ten million dollar bills, but he can readily get food from the underground free market, only the most fanatical will go to war against the underground free market.
Pro Christo et Libertate integre!