Assuming $P$ is the transfer function of your process and $C$ the transfer function of your controller.

The closed-loop transfer function of a standard control loop, with the controller in the normal path, is

$$ G(s) = \frac{C(s)P(s)}{1+C(s)P(s)}$$

while the transfer function of your alternative controller is $$G_{alt}(s) = \frac{P(s)}{1+C_{alt}(s)P(s)} $$

If you set your new alternative to controller to

$$ C_{alt}(s) = \frac{1+P(s)C(s)-C(s)}{P(s)C(s)} $$

You will get the same closed-loop transfer function (G(s))

That being said, not sure it's a good idea for 2 reasons

1 - To get to same result as the original controller, you need to cancel pole and zeros in the alternative closed-looped transfer function. If you cancel a RHP zero with a RHP pole, your final transfer function will seem stable however your controller will not be, the output of your alternative controller could become unbounded.

2 - Even if you don't cancel RHP zeros with RHP poles, this method is way more complicated.

Assuming $P$ is the transfer function of your process and $C$ the transfer function of your controller.

The closed-loop transfer function of a standard control loop, with the controller in the normal path, is

$$ G(s) = \frac{C(s)P(s)}{1+C(s)P(s)}$$

while the transfer function of your alternative controller is $$G_{alt}(s) = \frac{P(s)}{1+C_{alt}(s)P(s)} $$

If you set your new alternative to controller to

$$ C_{alt}(s) = \frac{1+P(s)C(s)-C(s)}{P(s)C(s)} $$

You will get the same closed-loop transfer function (G(s))

That being said, not sure it's a good idea for 2 reasons

1 - To get to same result as the original controller, you need to cancel poles and zeros in the alternative closed-looped transfer function. If you cancel a RHP zero with a RHP pole, your final transfer function will seem stable however your controller will not be, the output of your alternative controller could become unbounded.

2 - Even if you don't cancel RHP zeros with RHP poles, this method is way more complicated.