Over Paneling, Over Inverter Voltage with Breaker

[Ampster]

Could I simply add a breaker at 250v for each string right before the inverter? Any cons to this other than the breaker failing and not tripping?

Yes, if the breaker takes a few milliseconds longer than it takes your charge controller to release the smoke you will have a costly issue. As others have already mentioned, I am not even sure there are breakers but there are voltage operated relays. How accurate and how fast they are may vary. There is no negative to design for safe voltage, especially since you can over panel within safe voltage and not worry.