The fine folks at Octopus Deploy pointed me to this help page that got me most of the way there.
For anyone else who runs into this particular problem, I originally added this to my .nuspec file:
<files>
<file src="bin\x86\*.*" target="bin\x86" />
<file src="bin\x64\*.*" target="bin\x64" />
</files>
but nothing got copied; so, I changed it to this:
<files>
<file src="bin\x86\SQLite.interop.dll" target="bin\x86" />
<file src="bin\x64\SQLite.interop.dll" target="bin\x64" />
</files>
Then TeamCity had a build error because x86 and x64 were empty. It looks like OctoPack somehow runs before those files get copied. It's a hack that I hope to remove at some point, but I got things working by adding those two files to my project, and changing my nuspec file to this:
<files>
<file src="SQLiteFiles\x86\SQLite.interop.dll" target="bin\x86" />
<file src="SQLiteFiles\x64\SQLite.interop.dll" target="bin\x64" />
</files>
Also, don't forget to add OctoPackEnforceAddingFiles=true in TeamCity.