I would advise you to book all travel on one ticket.

If you travel on separate tickets, you risk being refused travel by the first airline on your ticket to DUB, even if you show them your onward separate ticket.

Travelling on separate tickets is never advisable when there are potential visa issues. Whether rightly or wrongly, when you present an air ticket showing your destination is Ireland, you risk being refused if you do not have the travel documents (other than an onward ticket!) acceptable to enter the country.

You can also most definitely buy an Avios ticket that covers both journeys, even if it requires calling to book it