Earliest preferred POD is 1360, after the Black Death basically
The challenge is to achieve an Ireland under Britain were a large majority of the population, let's say 80% at least, identifies itself as British. I don't require it to be the only cultural-political identity of the island, as of course Irishness would remain a concept in most scenarios by virtue of geography.
What is not allowed or not preferred is industrial population movements, mass genocide and stuff like this, having stuff like the actual Ulster plantations, massacres that are not widespread and that are limited to warfare, emigration directly or indirectly caused by outside forces and stuff like this is allowed, although I doubt you could achieve the challenge by having a timeline were the later events are more present than OTL, but you can challenge me on that.