What payment gateway should be used for invoice payment?
Yes - just create a new (custom, standard) payment method named Invoice, and user's then can select that.
On the payment method edit form you can add more details, and even could limit the availability of the method.
Rules are not actively supported as in D7: https://www.drupal.org/node/1535554
But I do not see a need using rules with your use-case.
You need to configure your invoice type to send emails, there should be a checkbox: Email the customer a confirmation when an invoice is generated - see https://www.drupal.org/project/commerce_invoice/issues/3089861
The invoice is attached by default and you can remove the attachment with hook_mail_alter, it's using the id invoice_confirmation. See https://git.drupalcode.org/project/commerce_invoice/-/blob/8.x-2.x/src/Mail/InvoiceConfirmationMail.php (UPDATE: The invoice types seem to create invoices for all payment methods and is not (yet) configurable to limit them)
See also the commerce-invoice-confirmation.html.twig to change the email.
Not attaching the PDF makes the emails smaller, and maybe passes more SPAM filters .. But users need to take action and press a link .. I would discuss that with the customer or maybe try both and gain experience with them. (Like what method is really paid more, and which one is faster)
UPDATE:
If you only want to send the invoice for one payment method you need to alter the commerce receipt email (id: order_receipt) and attach the invoice there. You'll find the order entity in the body's theme render array (first entry using #theme commerce_order_receipt)