You can use the experimental (?{...}) code groups in the generated regex:

use strict;
use warnings;
use 5.010;
use re 'eval';

my %h = (
   '^foo' => sub { say 'foo' },
   '^bar' => sub { say 'bar' },
);

my $what;
my $re = join '|', map { "$_(?{\$what = '$_'})" } keys %h;

if ('foo' =~ m/$re/o) {
    $h{$what}->();

}
[download]

But be sure to read the warnings in perlre.

Note that this requires the keys not to contain ' or \, but you could easily give them identifier labels instead, or properly escape $_ before interpolation.

(5.010 only required for say).