I just found out about someone else also working on SVG background demos with impressive results! On this forum thread, Holger Will has been demonstrating his attempts, which include rounded corners.
After a quick look at his SVG files, it seems he solved the stroke problem basically by drawing each corner as filled shapes, rather than use stroke at all. Frankly, this makes too much sense and I can’t believe I didn’t think of doing that. The one drawback is more complicated code (it would appear more difficult to modify than my method), but the effect is certainly worth it.