Code-wise, the third option is the easiest to implement, at least if you pre-initialize your key-hash with a default:

my %values= (
    '^' => '^',
);

...

s!\^\(([^)]+)\)!$values{ $1 }!g;
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Due to the selection of a default key of ^ with value ^, ^(^) gets replaced by ^ without any change to your code.

Of course, this approach isn't immediately applicable to your situation because you seem to want to dispatch method calls. A really nasty approach here would be to implement a ^ method in your object that returns the value ^, but that brings us deep into obfuscated territory, and the superficial simplicity of the code will certainly mystify the casual onlooker:

package My::Document;

*{"My::Document::^"}= sub { "^" };

sub name { $_[0]->{name} };

sub fill_template {
    my( $self, $template )= @_;
    $template=~ s!\^\(([^)]+)\)!$self->$1()!ge;
    $template
};

package main;
my $t= bless {
    name => 'wanna_code_perl',
} => My::Document;

print $t->fill_template(<<TMPL);
    Hello ^(name).
    Escaped: ^(^).
TMPL
;
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