Your working command includes backticks while your Perl string does not contain them.

If you want to send backticks to the remote side, you will need to include them in the string, and hope that you get all the shell quoting correct. Personally, I prefer to use $() shell-syntax instead of backticks as that makes the quoting much easier. Instead of backticks in Perl, I will use qx(). That gives me

my $command = qq[$SSH root\@$one for i in \$(lspv | awk '{print \$1}')
+; do bootinfo -s \$i; done | awk '{sum+=\$1}END{print (sum\/1024)}'];
warn "[[$command]]";
my $space  = qx($command)
[download]

I don't know if I've translated your shell code properly, as to me the original seems to contain a lone double quote and some spurious single quotes.