Hi Matthew,
I tested this at Pro 2.5 and I'm not seeing the issue you describe.
Map on the left is a polygon map that has had the frame rotated on the layout - not the interior map rotated
Map on the right is a rectangular map that has the frame rotated on the layout - not the interior map rotated
the labels for partial features are displaying for both.
for you at Pro 2.4.2 it sounds like you're just trying to force the labels over so they don't conflict with a layout element you're placing overtop of the map frame.
What you can try is in the map create a polygon feature class and create a polygon that is the same shape and location as the element you're placing overtop in the layout. then make that feature invisible by giving it no colour.
if you then increase the weight of it in the labeling pane it will not allow labels to place over top of it. giving the impression that they're avoiding the layout element.
hope that helps,
Wendy