Oh, I understand now.
Before I start trying to reproduce your problem, I don't understand your values. When I select the white filled/black border round corner rectangle on Border layer, it's 239.286 x 333.571 px.
Oh, I see, you're exporting at 300 dpi, and you're getting your values from the Export PNG dialog, rather than the Selection tool control bar. I guess you must need the higher dpi for the printing? But using 300 dpi for these "digital" versions means that they will be bigger than you drew them. So you might want to check the final size of the exports.
Ok, the reason why using export selection seems to work on the white card, is because the background is white/transparent, and the export "engine" (for lack of the proper technical terminology) "sees" and captures the black border. But the exported image is still rectangular. The image itself does not have rounded corners.
It doesn't seem to work on the black one, because the Selection option for Export PNG is only for the bounding box, which is always rectangular (or square). The Selection option does not use the actual path which is selected, within which to export.
Using the Selection option is a Great trick under certain circumstances, but it's always rectangular.
You should be able to clip it, and export that image.
Yeah, I think you might not have the right idea of a mask. I don't see any reason to use a mask here. (A mask is very similar to a clip. The difference is that the mask usually has a gradient or filtered (blurred) area in the fill, and that gradient or filter creates a variable mask over top of the image. It's sort of like putting a cloud over it....it's kind of hard to describe....well, for me it is anyway.)
I think I see your problem with the sizes being different. When you clip something, the stroke is irrelevant. It clips to the raw path, without the stroke. So the white one's size, because it was created with Export PNG, it includes the stroke width in its size.
So you just need to make a clipping path of the proper size (without the stroke). And rather than nudging it around, you can center it over the black/starfield background with 2 quick steps.
1 - select the clipping path first
2 - add the background to the selection (use Shift key)
3 - Object menu > Align and Distribute
4 - in the Relative To: dropdown menu, choose Last Selected
5 - click these 2 buttons in any order
Well, I guess it's more like a few quick steps, but still less tedious than nudging.
Sort of off topic, but I wanted to say that I think you really did a great job with that book on fire drawing. Very effective!