Using Bit::Vector.

use Modern::Perl;
use Bit::Vector;
use Data::Dump qw/dump/;

my %conversion = (
    ' ' => 0,
    0   => 1,
    1   => 2,
    2   => 3,
    3   => 4,
    4   => 5,
    5   => 6,
    6   => 7,
    7   => 8,
    8   => 9,
    9   => 10,
    A   => 11,
    B   => 12,
    C   => 13,
    D   => 14,
    E   => 15,
    F   => 16,
    G   => 17,
    H   => 18,
    I   => 19,
    J   => 20,
    K   => 21,
    L   => 22,
    M   => 23,
    N   => 24,
    O   => 25,
    P   => 26,
    Q   => 27,
    R   => 28,
    S   => 29,
    T   => 30,
    U   => 31,
    V   => 32,
    W   => 33,
    X   => 34,
    Y   => 35,
    Z   => 36,
);
my %inverted;

my $vector = Bit::Vector->new(6);
for ( keys %conversion ) {
    $vector->from_Dec( $conversion{$_} );
    $conversion{$_} = $vector->to_Bin();
    $inverted{ $vector->to_Bin() } = $_;
}
while (<DATA>) {
    chomp;
    my $string = join '', map { $conversion{$_} } split //;
    my $vector = Bit::Vector->new_Bin( 36, $string );
    my $bits = $vector->to_Bin;
    $bits =~ s/([01]{6})/$inverted{$1}/ge;
    say $bits;
}

__DATA__
123456

ZZZZZZ
ABCDEF
[download]

I know, Bit::Vector is an XS-module, so it doesn't comply with your specs.

On my old laptop, it reads a file of 1 million strings, converts them to the compacted format and reconverts them to the original format (to make sure it worked) in 100 seconds.