I have fixed the practical problem, in the mean time, by (a) removing the bottom copper layer in the SVG and (b) setting the part to SMD in the connection editor. I'm unsure which of these fixed it (or maybe both are needed?).
But in my opinion it's really a bug that the autorouter thinks the top and bottom layer are connected (without me providing a via), whereas the fab process doesn't connect them.
It'd be nice to have some real documentation on the file format and the intricacies of the multilayer SVG files and how they're (presumably) converted into multiple SVG files internally. What exists glosses over details (specifically for niche areas like SMD components).