Bitpay isn't the most Bitcoin friendly service. If you take custody of your BTC with a wallet that allows you to set the fee amount and consolidate your inputs during low transaction periods you will not have such an issue with high fees.
I never realised this was possible. Which wallet service?
Wallet service? I said, "If you take custody of your BTC" meaning that you would use a wallet on your own and not an online service. It's as simple as sending yourself a transaction with all your small inputs when you can confirm it on a block with a low fee.