The way the shipping charge rules are designed you should have a shipping charge that is equal to the quote returned from USPS. If you can see the full response and the amount differs from the charge you see in the cart then the problem may not be the API interaction that is occurring.
To check this take a look at your shipping rule and ensure that it is set to just use the value of the quote for the charge. If it is setup this way then we will need to look more closely at the api interaction and where the values are going. Please confirm the details of your shipping rule and if needed I will open a ticket on this so we can investigate it further.