mail-relay.3ireland.ie should work.
I consider this a very bad situation, where you can only send emails through one portal. If they have a problem it leaves you unable to send email at all, which is unacceptable.
Your provider has deliberately crippled your internet access, and yet it's allowed. I have several providers and have to change my email SMTP settings in order to sastify each one. That's not good enough. 3 are the worst as they insist that all emails are sent through their servers, .. that's terrible... a total bottleneck., and it does represent a deliberate crippling of the internet by Three, which isn't mentioned in any of Three's literture.