i really think you are underestimating the value of modern radars and sonars. As shown here, even a simple modern airship can totally outmatch the anti-air weaponry of the era, and as im sure you know, information is vital in war.
Even if they do run out of missiles, all the have to do is stay in the back and use their superior scouting sistems to stay a step ahead of the US navy, while using the imperial navy as a shield.
Granted, they really dont want to do that cause it would chance the wars outcome, but they still do it can if they need to.
Remember that Kadomatsu has always said that their first priority is to keep the mirai safe, and second is to save people, so if bad comes to worse, they will switch tactics to keep themselves alive.