Answer
This live, pre-save refresh isn't achievable — a read-only custom attribute's underlying logic only executes once the record is actually saved, so there's no way to make it populate reactively in response to an in-progress selection before that save happens. Where showing the related value earlier genuinely matters for usability, adding that same underlying value into the List-of-Values/lookup dialog itself (so users see it while selecting, rather than after) is a practical alternative to a live client-side refresh.