The issue is that nobody would ever do this because it's against their own interests. If one ISP charges per byte and you offer to pay the charge, you've deprived that ISP's customers of the incentive to switch to an ISP that doesn't do that (which you then wouldn't have to pay for) and given every ISP that doesn't do that an incentive to start, so that you have to pay for them too. And then the ISPs can raise their fees without having to worry about losing customers because the people paying aren't the people deciding which ISP to use. It's economic suicide. Which is why it's rational to presume that anybody who suggests it is in bed with the ISP, which is then the obvious antitrust problem.
But let's just put the final nail in the coffin here. Does the music service want to pay their customers $1 for every gigabyte of music their customers listen to? Fine, pay it regardless of what the customer's ISP charges for bandwidth, the same amount to any customer on any ISP. If doing that is profitable then it's profitable regardless of what the ISP charges the customer, it's better for the music service because it doesn't give the ISP any pricing leverage on bandwidth charges, and if paying the customer money to encourage them to use the service works for customers with per-byte billing then it should work just as well for customers with unlimited plans, so the proposal will earn them even more customers.
But let's just put the final nail in the coffin here. Does the music service want to pay their customers $1 for every gigabyte of music their customers listen to? Fine, pay it regardless of what the customer's ISP charges for bandwidth, the same amount to any customer on any ISP. If doing that is profitable then it's profitable regardless of what the ISP charges the customer, it's better for the music service because it doesn't give the ISP any pricing leverage on bandwidth charges, and if paying the customer money to encourage them to use the service works for customers with per-byte billing then it should work just as well for customers with unlimited plans, so the proposal will earn them even more customers.