Nope unless you ask them to upload it via the SVG code and add an input for that. This method is far from being user friendly, like the solution you said aswell.
What I've done previously is I've set a link next to it where they can upload (on my server) the svg and then I show them the SVG code and copy it to the clipboard. Still far from ideal, but makes it easier to understand.
Hope they will bring in future update, section update was one of the best update for theme settings lots of flexibility.