Symbol not found: _xmlFree

That's the killer right there.
LibXML.xs calls xmlFree(), expecting that function to be resolved by your libxml2 library, but it's not.

Either your libxml2 library does not contain that expected functionality (which I think is unlikely), or your libxml2 library is simply not being located and loaded at runtime.
I don't know exactly what would be required to fix this on your system.
I think that, in building XML::LibXML-2.0206, there are some pre-tests run to check that libxml2 is available - so it's a bit odd that this problem is not detected at the beginning of the build.
Perhaps those pre-build checks don't extend to verifying that the libxml2 dynamic library is going to be found when needed at runtime ... I don't know.

I did try to force the install, and it installed, but the perl script I'm trying to run still came up with the same error, so I don't think it was successful

Well ... it installed successfully ... but the underlying problem of the failure to resolve _xmlFree at runtime has not been addressed.

Cheers,
Rob