Never mind - I figured it out, more or less:
using SKTextBlob sKTextBlob = SKTextBlob.CreatePathPositioned(LabelText, font, LabelPath, LabelPaint.TextAlign, new SKPoint(0, 0));
SKRect boundsRect = sKTextBlob.Bounds;
The only slightly tricky part was creating a SKFont from a Windows system.drawing.Font.
The resulting bounding box is somewhat larger than the text (it's a conservative bounding box, according to the documentation), but it works for my purposes. I don't need a tight bounding box for what I need to do.
Thanks to anyone that looked at my original post.