I'm not seeing any good to come of taking a coin INTO an archaeological site unless you relish having to prove that you brought it in rather than found it there. You most certainly should have proof of prior legal ownership with you. This reminds me of a problem some photographers had returning to the US from an overseas trip with their Japanese/German cameras and being expected to pay import duties again. For that matter, has anyone sold a coin to someone in Italy, for example, and had it returned? You might be able to prove the status of the item but why set yourself up to have to?
While I suppose this could conceivably be an inconvenience in some jurisdictions, we’re talking the UK here. To my knowledge they have most enlightened antiquities regulations in the world. No government agents lying in wait to entrap a tourist with an ancient coin in their possession. I imagine if I had a metal detector and shovel things could have become interesting, but a coin in a leather pouch. Never gave it a thought.
I really wish I did. I regret not doing so. It was wild, the object was just hanging out on the ground, in plain sight. See those stacked globular things on the left and right? It was a baseball/softball sized broken off part of one of those things.
If it’s history you got to give it up. If you have a whole nation looking at it. Plus your name on it