I've not read the briefs in Miller, but I would expect the prosecution would have made a prima facie case that sawed-off shotguns weren't "militia" weapons. If not, they certainly could have done so easily. The burden would then have been on Miller/Layton to rip that prima facie case to shreds, which they too could have done easily.
I don't think that the lower court would be required to accept a light burden of proof. Unlike the Miller Court, the lower court made the right decision to begin with. Even with an invented "suitability" test, I could imagine the lower court dismissing again for lack of evidence. What evidence would convince such a lower court? The need for close quarters with a sawed-off shotgun is no more limiting than that needed for use of a bayonet.