Hm, I suppose it could go without being secret but it would be a bit more interesting otherwise.
And yes, the region lord pays or gains the difference in troop prices.
As long as we think of a way to not just create a money printing machine, where lords set the cost to 0 and just accumulate thousands of golds' of debt.
On a tangent, perhaps consider potential applications of the war chests to recruitment instead of just repairs. A return to a more collectivistic approach would be beneficial, in my opinion. The social changes combined with the decrease in density have made waging wars a lot harder than they used to be, with the result of them becoming more infrequent in at least many places.