I would hope that Bitpay should forward that amount to you without charging you for a "network fee"... in my experience as a user (ie. buying things from merchants who use Bitpay as their payment processor), they actually add an extra network transaction fee to your purchase amount to cover the cost of Bitpay consolidating all the payments and transferring to the merchant.
For example, Buying an Amazon Gift Card from eGifter.com:
$100 AGC = 0.009150 BTC
Bitpay invoice = 0.009150 BTC (card) + 0.00026000 BTC (network fee) = 0.00941000 BTC total.
You then need to put ANOTHER network fee to send that total amount to Bitpay from your wallet... You can't control the transaction fee that Bitpay charge, but (depending on your wallet) you can control the transaction fee for sending the transaction TO Bitpay
As I understand it, Bitpay then use the extra fee paid by the purchaser to cover the cost of transferring the coins to the merchant. According to their pricing (
https://bitpay.com/pricing) they may also take a slice of the 0.00915 purchase price (up to 1% as a service fee depending on your plan/transaction volume)...