[Finn Haverkamp] “When editing the sequence, I can right-click the 4K clip and “Set to Frame Size” or “Scale to Frame Size”, but I think that’s just visual for the preview.”
That’s how you scale it, and it is not just for viewing, it will reflect in your rendering as well. If you want it to happen automatically, then go to your Premiere preferences menu option under General will be a checkbox for “Default scale to frame size”. If you check that and then click “Ok”, then Premiere will automatically downscale your 4K to show the full clip in your 1080 sequence.