If you're happy to coerce from an arrayref ([$dsn,$u,$p,\%opts]) instead of a hashref, then:

package Moo::Role::DBIConnection {
    use Moo::Role;
    use DBI;
    use Types::DBI;
    has dbh => (is => 'ro', isa => Dbh, required => 1, coerce => 1);
};
[download]

Coercion from HashRef isn't provided in Types::DBI (because that would require choosing hash key names, something probably application-specific) but it's pretty easy to add in if you need it:

package Moo::Role::DBIConnection {
    use Moo::Role;
    use DBI;
    use Types::DBI;
    use Types::Standard qw(HashRef);
    has dbh => (
        is         => 'ro',
        isa        => Dbh->plus_coercions(HashRef, sub { DBI->connect(
+@{$_}{qw/dsn user password options/}) }),
        required   => 1,
        coerce     => 1,
    );
};
[download]