In my opinion, the fraction of the cost of the army that goes to preserving fiat is somewhere in the region of 0%.
What maintains the value of fiat is not government coercive action (e.g., having an army that forces people to use currency). Instead, the value is dictated by how much the populace believes it is worth. So it's the general government actions to maintain the health of the economy, and none of that would disappear if the government were to decide not to use fiat.
Indeed, to the extent that you'd compare how much the government spends in a hypothetical world where it doesn't use fiat currency versus one that it does, it probably takes more resources in the first case (i.e., fiat currency is a net asset to the government, not a net cost).
What maintains the value of fiat is not government coercive action (e.g., having an army that forces people to use currency). Instead, the value is dictated by how much the populace believes it is worth. So it's the general government actions to maintain the health of the economy, and none of that would disappear if the government were to decide not to use fiat.
Indeed, to the extent that you'd compare how much the government spends in a hypothetical world where it doesn't use fiat currency versus one that it does, it probably takes more resources in the first case (i.e., fiat currency is a net asset to the government, not a net cost).