Current controlled bridge amplifier
So if I understand your question correctly, you don't want to regulate the voltage at all, but you want to regulate the current. So just take your output current and turn it into a voltage and put feedback around the amplifier to make it match a reference voltage that you scale proportional to the current.
I used your load resistor in this example, but you could just as easily add a sense resistor.
Depending on your swings, you'd need to bias R12 to halfway voltage maybe.