Nope, you can do a SODA in a C-172. I am not sure if you can get a first class medical if you are completely deaf on one ear but there is a big chanse that you can. Best thing is to contact the FAA medical devision and ask about hearing and SODA. Maybe the can issue you a medical certificate with restrictions. For example, valid only for multi-crew operations or not valid for passenger operations (only freight), maybe they will allow you to only become a flight instructor etc. Best thing is to contact the FAA.
In my flightschool there was an ex Navy pilot that crashed in to a Carrier with an F-14 and he got spine injury that resulted in not being able to walk.
He bought himself a Cessna 402 and did some modification to it so he did not need to use the rudders with the feet. He did his SODA with his aircraft at our school. He received his medical and was able to start his own freight company.
The FAA are the most flexible and logical authourity in the world without comprimising safety.