I assume you haven't had any responses because nobody knows of a feature that does what you want automatically. FWIW, I don't see any lines at the top of the second column.

I note that Word doesn't have a way of automatically putting the repeated heading at the top of the new column either (unless your content is in a table). Since you need to manually insert this second instance I would just have an alternate style to apply to the second instance of the heading.

I would suggest a simpler alternative that removes the messiness. If the styles of both artist and song include the paragraph setting to 'keep with next' and you precede each artist paragraph with an empty paragraph that has a tiny line height and does not 'keep with next' then you would avoid any of these flow-over instances happening unless an artist had more than a column's worth of songs.

If you are using tables to get the heading to repeat automatically, you might be able to find relevant threads where people have asked how to include a (continued) on subsequent pages. This can be done with a floating white box but is so complicated that most people don't bother unless the requirement is VERY IMPORTANT.