Welcome to SolidWorks forums Aagens.
This is a current limitation i.e getting the surface area of surface body under file properties. Please vote for SPR 393916.
At the moment you can try/use this workaround. Create a property named Area or Surface Area in the part file properties. Now using the measure tool select all surfaces/faces and get the area. Copy and paste that value under file properties for the area property. Repeat this for all the parts. Now in the BOM add a new column and link that to the new property. Disadvantage of the system is that on any change, the area needs to be checked and updated manually.
I'm not sure if this can be automated via a macro but will check.
You might also find this post helpful for your future reference.