Totally discreet, no need to use your real name, go in as a guest and call yourself what you want. Everyone else does. Just make sure you either destroy your return voucher or keep it safe from prying eyes. Many a guy has fallen at that hurdle.
As for when it is busy? The person that could predict that could take over the Mystic Meg column. You pay your money and take your chances. No good ringing up to find out, the staff do not give out that information and it would only be correct at that moment anyway.
Good luck