To address Gaff's point, it was noted that you want four boxes. It was also noted that you have to address a range of venues and applications and that you currently have four mains. That made it seem reasonable to assume the four boxes were four mains. But then bass and keys were noted and when a couple of two mains and two subs options were suggested it was commented that those were what you had been considering. So that makes it seems that the four boxes would be two full mains and two subs. It was noted that separate subs are acceptable but also that setup and operations needs to be as fool proof as possible and there were references to using just two speaker for some applications that would seem to call for full range boxes. So what is actually desired in terms of the types and quantities of speakers is not real clear, and may not be for your either.
Easy things to overlook because they are probably blatantly obvious to you but we don't know the physical size of your school quad, or at least the area of it to be covered, or the size and acoustics of the stage, jazz ensemble classroom and feeder school cafeterias or the background noise levels you may have to overcome in the quad. or the audience sizes involved. Or the levels you run. That makes it difficult to answer questions such as whether it would be loud enough.
Another aspect that may be worth considering is just how much improvement you expect and what may be achieved. For a quick example, let's compare your existing Peavey Impulse 200 to the QSC K12. The maximum overall output is essentially the same, 132dB@1m for the Peaveys versus 131dB@1m for the K12s. The Peavey has a wider horizontal and shallower vertical coverage, which when the speakers are used on stands is usually desirable for covering a larger plan area and/or in spaces like cafeterias that often have relatively low ceilings. The frequency response of the K12 is more extended. Basically, the K12 is most likely going to sound better but it may not get any louder and the coverage pattern may not be as desirable for the applications noted. So if the sound quality is the primary factor then there may be a significant improvement, especially if you add subs. However, if you wanted comparable coverage and/or a noticeable increase in potential overall output then the QSC K12 may not be a great choice.
Easy things to overlook because they are probably blatantly obvious to you but we don't know the physical size of your school quad, or at least the area of it to be covered, or the size and acoustics of the stage, jazz ensemble classroom and feeder school cafeterias or the background noise levels you may have to overcome in the quad. or the audience sizes involved. Or the levels you run. That makes it difficult to answer questions such as whether it would be loud enough.
Another aspect that may be worth considering is just how much improvement you expect and what may be achieved. For a quick example, let's compare your existing Peavey Impulse 200 to the QSC K12. The maximum overall output is essentially the same, 132dB@1m for the Peaveys versus 131dB@1m for the K12s. The Peavey has a wider horizontal and shallower vertical coverage, which when the speakers are used on stands is usually desirable for covering a larger plan area and/or in spaces like cafeterias that often have relatively low ceilings. The frequency response of the K12 is more extended. Basically, the K12 is most likely going to sound better but it may not get any louder and the coverage pattern may not be as desirable for the applications noted. So if the sound quality is the primary factor then there may be a significant improvement, especially if you add subs. However, if you wanted comparable coverage and/or a noticeable increase in potential overall output then the QSC K12 may not be a great choice.