I wouldn't recommend using dysfunction as a verb. Nobody else does, so you'd either come across as an idiosyncratic maverick, or a non-native speaker.
To a considerable extent, dysfunction/malfunction can be defined as synonyms, but in practice dysfunction tends to be used in medical/sociological contexts, while malfunction is more likely in technological contexts (plus it's the default for all other contexts).
I don't think there's any justification for OP's assumption that dysfunction somehow implies a more extreme form of malfunction. Okay - most people would agree a dysfunctional family is worse than a malfunctioning hard drive, but don't forget the Apollo 13 malfunction.
It's not obvious to me there's any real need to use dysfunction as a verb. Obviously I'm biased against OP's example usage simply because "that's not how we say it", but it also seems to me that there's something slightly "oxymoronic" about referring to a family collectively (so it can act as the subject of the verb) when semantically it's not a complete unit, acting as a single entity.
In short, English is indeed something of a "free-for-all" for native speakers in certain respects, but you really need to know the rules before you deliberately break them. And this context, that's in the sense of as a rule, not rule of law.