What might be nice is something which did both:

use bimodal_constant FOO => "BAR";

print "\$FOO is $FOO\n" if FOO;
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That seems as simple as:

package bimodal_constant;

sub import {
    my ($class, $key, $value) = @_;
    my $caller = caller();

    # constant scalar
    eval " *$caller\::$key = \\\"$value\" ";
    # constant sub
    eval " *$caller\::$key = sub () { \"$value\" } ";
}

1;
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When used with the example at the top, it prints "$FOO is BAR" as expected.

Of course, a shorter name would be helpful.

Update: glancing at the source for constant, there is a lot of sanity checking so it's not quite that easy, but the sanity checks could be cut and paste. All the action is in the last few lines of code.