As Safety Concerns has stated quite correctly, you need the correct engine/airframe combination type rating in order to certify that aircraft at all.
If you have just V2500 you cannot certify a landing light SBU change on a CFM'd 320 because you do not have the type rating, to do so would be illegal.
The fact you have CFM on your 737 makes no difference at all and will NOT get you any kind of type rating issue or 'limited approval' on that aircraft. As you have stated the Airbus uses a CFM56-5 and the Boeing a CFM56-7.
You'll need to go and do a course to have the CFM56 put on your license, along with the relevant Practical Training/OJT (none of your B737 sheets will count!).
Don't forget it isn't just an engine course; it's an Engine & Airframe Interfaces course.