OK I actually think I just found the solution with a bit of experimenting.
I added a Lerp node and put the emessive color on A and the texture of the wood material (the one that isn’t supposed to have an emission) on B.
That actually adds a little bit of emession to the B texture too as expected but there was a simple solution to that.
I added a Multiply and a Constant node to B before I connected it to the Lerp and it removed the emission from the wood material.
It may look like a mess but this is the final result.