Shops are normally quite good as these days as they want to keep good customer relations.
If the top is only 3 weeks old then I'm sure they'll still have it in stock so should recognise it.
However as you haven't got a receipt then they normally will give you vouchers or a gift card to use in store.
Or if you want it again then just exchange it.
Also if you want a refund and it is now reduced or on sale, then they will only give you the lowest amount, as you don't have any proof they may think you've just brought it cheap and now trying to get a full refund.
have you checked the washing guidelines? Some with jewels on say handwash only. If not don't worry, if it does just say that it fell off, otherwise they will say it's your own fault for not reading instructions.