While there may be a better way to do this, I just added a simple regex:

#!/usr/local/bin/perl
use strict;

my $name = "Fred";

my @infile = <DATA>;
my $tmpl = join '', @infile;

$tmpl =~ s/\$name/$name/g;

# output
print $tmpl;

__DATA__
This is my template.
Here is my $name
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