Omega 3 enriched generally means they're fed a vegetarian soy and corn based diet with additional supplementation in the form of fish or flaxseed oil. Perhaps it's an hormonal reaction? I tend to have hormonal disturbance when I eat soy or flax, so it could be down to that for you. Do you have access to fully pastured eggs? I would try an elimination diet and see if there aren't any differences between the eggs you're currently eating and the pastured alternative. Cattle in Ireland is predominantly grass-fed, but it's possible there might be some grain supplementation going on, you could probably contact the company/farm to find out for sure.

As for why it doesn't effect you when it's fully cooked, well... Not a clue :P But personally I think it's worth assessing the quality if it means you don't have to destroy all the deliciousness of a rare steak :)

My skin issues resolved after supplementation with vitamin A (which I took with Zinc for kicks and giggles and be careful to maintain around a 10:1 A to D ratio). This also cleared up my dairy intolerance which was sort of a neat trick.

Try to eat raw meat without the eggs and tell me your results. Blend in some liver pate or olive oil, if it's too dry for you. If i eat mono (some meat raw and cooked, some fruits, starches, no raw eggs, no veggies, no nuts) I've a better skin.