I created this code a few days ago. It's pretends to be the constant module and overrides constants to what you like. It's mostly a proof-of-concept but you can try it.

# file: constant.pm
package constant;

use warnings;
use strict;
use File::Spec;

my %Overrides;

sub make_inc_hook
{
    # Make sure we don't load this file... or we'll recurse infinitely
+ later.
    my ($path) = ( grep { -f $_ }
                   map { File::Spec->catfile( $_, 'constant.pm' ) }
                   grep { $_ ne q{.} } # avoids our directory
                   @INC )
        or die q{couldn't load constant.pm};

    open my $fh, '<', $path or die "open: $!";

    # Filter the real constant.pm, rename package to old_constant.
    my $source_filter = sub {
        return 0 unless defined $_; # edit: added 'defined'
        s/package constant;/package old_constant;/;
        return 1;
    };

    # Returns an @INC hook.  See perldoc -f require.
    return sub {
        my ($self, $file) = @_;
        return unless $file eq 'old_constant.pm';
        return ($fh, $source_filter);
    };
}

BEGIN { push @INC, make_inc_hook(); }

# Preload/compile the original constant.pm
use old_constant;

sub import
{
    my ($pkg) = caller;

    if ( ref $_[1] eq 'HASH' ) {
        my $param = $_[1];
        $Overrides{ $_ } = $param->{ $_ } foreach keys %{ $param };
        return;
    }

    my (undef, $name, $value) = @_;

    $value = $Overrides{ $name } if $Overrides{ $name };

    use Data::Dumper;
    local $Data::Dumper::Terse = 1;
    eval qq{package $pkg; use old_constant '$name' => } . Dumper( $val
+ue );
    die if $@;
    return;
}

1;
[download]

# file: A.pm
package A;
use strict;

use constant VALUE => 1;

sub func {
    print "Constant is ", VALUE, "\n";
}

1;
[download]

#!/usr/bin/perl
# file: example.pl

use warnings;
use strict;

use lib q{.}; # because constant.pm above is in the same dir...
use constant { VALUE => 1_000_000 }; # this overrides VALUE
use A;
A->func();
[download]

edit: Run 'perl example.pl' and the output is: Constant is 1000000