In looking at this a little more, this issue isn't with GeoGit, per se. The DLL causing this issue is being extracted from the sqlite-jdbc-spatialite-3.7.2-2.4.jar in GeoGit's repo directory. The real problem is that the DLLs for x86 and x64 are identical in that JAR, and that one DLL isn't x64, hence the error. There are x64 SpatiaLite libraries available for download, but those don't include JDBC DLLs.
As best I can understand it, my problem seems to be at the interface between Java and SpatiaLite. The
GeoTools JDBC SpatiaLite Plugin page states "Windows 64-bit libraries are not available but the 32-bit libraries should work on 64-bit systems." Maybe this is fundamentally my problem, i.e., the 32-bit libraries only work when I run GeoGit with 32-bit Java, so trying to go purely 64-bit may not be feasible, or at least practical.
Since 32-bit works, I will stick with that for now. I would rather spend my time exploring the potential of GeoGit than banging my head against a JDBC SpatiaLite issue.