On one hand I believe it's her fault for supplying you with the wrong address, but I know for a fact that if I was in your position, I would pay the postage myself. The reason for this is because it is easy to forget to update your address if you buy from many places. Paypal especially is easily forgotten, because anywhere you use Paypal, you probably have to put your address into the website you're buying from anyway, like here on Etsy, so people who ship to Paypal addresses are more likely to get it wrong than with Etsy addresses.
It was a mistake on their part, yes, but at the same time, it is your responsibility that they receive their parcel. It doesn't make much difference in my mind what the postage cost is, either. If it's £1, then it's too little to bother them over, and it's something I could easily afford myself. If it was £10, then I don't want to pay that, because I couldn't, but the customer really wouldn't be pleased having to pay double shipping. If it was the latter, though, I'd probably request half of it since it's expensive and their own fault.
$4, however, I don't believe is enough to bother them over, but it depends on your products and profit margin.