TheBearisBack my apologies but sometimes over the internet it is hard to figure out what’s behind a post.
Since this is not a beis midrash but, after all, a coffeeroom, I’ll take the liberty to write. I am female, I go shopping 🙂 I was taught that even people who have deities etc. normally we don’t treat like ovdei avodah zarah as long as they have some idea (possibly very distorted) that good is different from bad and that good is preferable to bad. Thus, if we see a pagan nazi y”sh etc etc, but for just about everyone else (hindu, catholics etc) we stop, try to help and call 911 even in the middle of nowhere. For a mistake like that or extra change, and where the mistake is absolutely not ours, I believe we keep and say nothing. We can return the extra item or extra money or offer to pay the lid, if doing so gives monetary gain (e.g. expected discounts in future) or good-name gain which is larger than the price of the lid (or extra change, or whatever). I believe it is assur to return the lid to the nonjew, regardless of their religion or lack thereof, because it makes us feel good to do so.