My answer would be:
The word dukkha is used with a few different senses throughout the suttas. Sometimes it means pain, as the opposite of pleasure. That’s mostly a “worldly” sense, the conventional use of the word. But sometimes it means anything that is not parinibbana, anything that is not the complete end of suffering. This is how the Buddha reemployed the word.
Suffering due to impermanence is not merely the fact that impermanent things inevitably change into some painful type of suffering; it is also that everything that is impermanent is not nibbana, and hence not the end of suffering.
So when the Buddha asks: “What is impermanent, is that suffering?” he is not just asking a psychological question. It is also an ontological one, namely that anything that’s not nibbana (anything that’s impermanent) is by that very fact suffering.
You could say this is just a matter of definition, but it isn’t. This is also how it is experienced by noble ones. It is only by contrasting the aggregates to the cessation of the aggregates that this scope of suffering can be understood, hence only noble ones truly understand dukkha. What others think is happiness, the noble ones know to be suffering, it is said in the sutta, and this includes the highest realms of existence.
(Dukkha doesn’t mean ‘unsatisfactory’. There’s many reasons for that, and I don’t want to derail the topic, but for one, it is the opposite of sukha, which doesn’t mean ‘satisfactory’ but ‘pleasant’.)