Gift Aid was one of the questions I asked my UK/US tax advisor about in my last catch up with them about a month ago. For what it's worth, he basically aligns with durhamlad. Paraphrasing, the advice I got was 'If you want to donate to charity in a US/UK manner, the best way is to make a single large donation to one that operates in both countries' - which doesn't really fit with the idea/purpose of gift aid.
I should point out in passing that many organizations (Oxfam, Red Cross, Comic Relief, etc) are actually legally and organizationally separate (but affiliated) organizations in each country they operate in. Meaning the organization you donate to in the UK may not be registered charity in the US despite nominally operating in both. Oxfam's Wiki page (
https://en.wikipedia.org/wiki/Oxfam) for example shows the 21 affiliated organizations and the respective countries they operate in.
I didn't ask my tax advisor (as I have FTC), but I think if you aren't claiming any FTC (or are claiming FEIE but are under the exclusion limit) gift aid wouldn't be problematic.
My practical workaround for donations to charity shops was to give them to a (tax paying) relative to donate. My workaround for entrance fees to places like Slimbridge Wildlife Trust is to allow someone else to pay my share, then pay them back.
Don't get me started on the Co-op card.