AnPost (the Irish Post) will be handled by USPS in the USA. The problem with AnPost is that the insurable value is very limited (€ 35 max.) on all non-EU mail. (Even within the EU the max. is only €150.) So the only failproof method would be FedEx or any other courier service with comprehensive insurance.
Although registered mail seldom gets lost, but occasionally it does.
Just had a recent case when my bag disappeared for one full month in the black hole of ISC NY (sorting / customs facility) with zero update - thankfully the buyer (a PFer) was very nice and patient about it. Then it moved on / got delivered just a few days before ebay's INR case-window was up. But i had some packets lost forever to ISC NY.
I think your seller is right about insisting on FedEx, it's too risky without proper insurance. Or if she's anywhere near to Northern Ireland, she could drive up there to avail of Royal Mail's better insurance (up to £500) - that's what i usually do when mailing pricier items overseas. Royal Mail also gets handled by USPS in the USA.