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>..to be instantaneous...

By which frame of reference? I never understood the language around "instantaneous". Isn't simultaneity relative? So that where one frame of reference says two events are simultaneous, there or others that say they are not?

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The old quantum theory was developed in non-relativistic setting, so this was not a concern. But you are right, relativity complicates lots of things in quantum theory, including the idea of "instantaneous" quantum jumps. In relativity, if some event is to be universally instantaneous, then it has to happen at a single point of space. Which is possible with point particles, but then you get the problem how those point particles can find each other to interact at a single point so often as measured cross sections indicate... perhaps they are not exactly points, but waves, but then we can't have instantaneous events, the event has to happen to the wave in big region of space where simultaneity is relative.

Simultaneity is only relative when events A and B are far enough apart that it would have been impossible for light to get from one to the other between the two events. When there is a single location, it's objective whether something has zero or nonzero duration.

And while observers disagree on their personal measurements of duration, they will always agree about what a clock sitting at the location will measure.

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