Other than adding more objects and another border (or moving them behind), is it possible to have the line cap option on just one end of a line?
I don't think that line caps are independent; just one line cap property for a Curve shape.
To create a rounded appearance on just one end of a Curve, you can create a custom arrowhead from a circular Ellipse.
Note that, if you do this, you may not be able to easily identify this arrowhead visually in the "picker". You might have to give it a descriptive name so that you can identify it by hovering over the picker.
I tried the arrowhead thing but couldn't get it to work right. They always turn out either too big or stick out too far. And some of these need to be changed thickness wise on the fly, so that would mean a custom arrowhead each time, if I understand how they work.
In my simple testing, it worked OK down to 1 point line width:
Whether that has practical benefit or not might depend on exactly what you need to do with the content. If I convert those outlines to objects, for example, the arrowhead way of implementing it produces "less perfect" geometry than doing the same thing with true rounded line caps - but I'm zooming in quite close to see that, and it might be perfectly acceptable for some applications.