It's easy enough to rig up your own tidy script with PPI that you can run after perltidy. Proof-of-concept:
use PPI;
my $doc = PPI::Document->new($ARGV[0]);
for my $op (@{$doc->find('PPI::Token::Operator')}) {
    if ($op eq '&&' || $op eq '||') {
        $op->{content} = " $op ";
    }
}
print $doc;
And if we run this script on itself, we get:
$ perl je8tidy.pl je8tidy.pl
use PPI;
my $doc = PPI::Document->new($ARGV[0]);
for my $op (@{$doc->find('PPI::Token::Operator')}) {
    if ($op eq '&&'  ||  $op eq '||') {
        $op->{content} = " $op ";
    }
}
print $doc;
It did insert the extra spaces around the only || operator on line 4.