The Royalty would be on the Gross amount.I'm inclined to think...

The Royalty would be on the Gross amount.
I'm inclined to think that the gross amount for hedged barrels would be the hedged price rather than the spot price.

On who pays the government, the short answer is that I do not know with any certainty.
My guess, is that purchaser (Shell?) payment to Byron would include the royalty, and Byron would pay the royalty to the government.
I don't believe that this would make a difference either way.

The purchaser removes a ($1.xx?) per barrel payment (as an alternative to interest payment).

I suspect that the purchaser would settle the full oil sale amount (less interest clip mentioned above) with Byron, and Byron would make loan repayments as per the agreed repayment schedule. These two payments could be synchronised, e.g., end-of-month 30 day invoice, and could be netted off.
Royalty payment may be on a similar cycle, but could be more in arrears.
Byron would be responsible for transport and processing costs.