Well, for now I solved my problem using SCP. The System SCP, that is, not the Perl Net::SCP module. I just happened to have tried that first and it worked for me. I changed all my code to use that, even the part of Net::SSH2 that worked for me (scp_put works just fine). At least it is all nice and uniform now. I talked to the developer of the module and I files a bug report on CPAN, following his advice.
Thanks,
Radu