International Shipping Charge Differs from Shopify Charge

We have had several instances where Shopify charges and collects an international client (in this case France) one rate for a USPS International First Class package and when we ship, we are charged a higher rate. There is no difference in package size or weight, just an overcharge of about $6 US on a $13 label. Needless to say we can not afford the overcharge and neither do we feel comfortable with going back to the customer for what is clearly Shopify’s issue. Any suggested fixes?