You don't want the quotes around BLAH.

#!/usr/bin/perl -w
use strict;

use constant B => 5;

my $name = B;

my $value = $name;

print "B gives: ", B, "\n\$name is: $name\n\$value is: $value\n";

__DATA_
B gives: 5
$name is: 5
$value is: 5
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Update From your followup I see I interpreted your question the wrong way.

With your code there is no benefit to avoiding no strict; for a small scope. You merely hide the issues.

Dispatch tables might suit your problem.