Your customer can't pay with a check for direct checkout, only credit cards or an Etsy gift card.
If they say the money has been sent to the bank, then it has been sent. It is now completely up to your bank how soon you get your money.
You did not even need to wait this long to ship. As soon as it is in the shop payment account, you are okay to ship. (f the credit card transaction hasn't gone through completely, the transaction will still be "processing", rather than "paid", and the money will not yet appear in your shop payment account.