tl;dr email support@topsec.com and ask to be whitelisted.

I have an email address using this domain. Every time I tried to email any Irish government org, I got:

This is the mail system at host mailout.nyi.internal.

I'm sorry to have to inform you that your message could not
be delivered to one or more recipients. It's attached below.

For further assistance, please send mail to postmaster.

If you do so, please include this problem report. You can
delete your own text from the attached returned message.

                   The mail system

<arts@whatever.ie>: host clustera.tstechnology.net[91.199.74.9] said: 554
    5.7.1 <arts@whatever.ie>: Recipient address rejected: [DNS_NAME] Sending
    domain hosted on bad dns server (in reply to RCPT TO command)

This is despite checking DKIM, SPF, etc - all the normal "don't flag me as spam" checks. It appears the Irish government's email vendor, https://www.topsec.com/, is blocking any email from a domain that uses Namecheap DNS, which is quite a few domains, unfortunately.

Emailing support@topsec.com from gmail got me whitelisted. I don't know another way around this.