Sarah: yes, birds in general are quite higly strung (to be colloquial) - they have heart attacks quite easily when their adrenaline is up. Not sure if that includes chickens, given they've been domesticated for quite a while, but rather safe than sorry.
Su, thank you for your detailed and helpful post.
So, I'm happy to report that the operation was a success (with my most grateful thanks to my partner and mother for their calming assistance).
We pinned her wings down as best we could with a rag (wrapped around her), then covered her head with a small cloth bag (big enough for her to breathe, but small enough to keep her head in the dark). We made sure to keep her legs in hand, as she was kicking around a little (my partner held the wings, my mother the feet, I was the worried surgeon).
I trimmed the feathers that were in the way of the wound and sprayed the area with Betadine (antiseptic spray). I used a razor blade (initially I tried a Stanley knife with a new blade, but that needed far too much pressure to cut the skin). I cut along where the stick was under the skin - which didn't cause much of a reaction from Pickle. However, when I tried to flush out some of the blood with the saline, she wiggled around a lot.
The stick was just under the skin, but it was attached around the entry point, where the skin had grown around it already. I cut around that as best I could, and was able to remove the entire thing. I flushed the wound again with saline - this time Pickle flipped out, lost her hood and nearly went off the table. We managed to calm her down again, and after a minute or so of keeping her still, we realised the bleeding was already slowing down markedly.
I specifically planned to operate an hour or so before sundown, so she'd basically spend only a very short time moving around before settling down to sleep and thus let the wound clot and heal a little. I'd set up a space for her that was clean, with food and water, away from the other chickens.
I put her in her temporary space, and 20 minutes later got called over by my partner to find that she'd escaped and was digging around in my garden, while trying to get to the other chickens (we only have four, including Pickle, who's the escapist of the group). So I let her into the coop, where she spent the hour or so before sundown acting as if nothing had happened. The wound isn't visible at all - it's covered a little by her breast feathers as well as being quite low on her chest.
She was up and running around the next day - as chatty and mobile as always, and the other chickens didn't seem to realise anything had happened. I picked her up to check the wound today (just over a day after the 'operation') and it seems ok. It's hard to spot - she's a wriggly chicken at the best of times and she's wary of me at the moment, rightly so.
Thanks again to everyone who weighed in with their knowledge and advice!
