Sorry, but no.
As I've mentioned before, Apple stupidly recycled the A1278 model number for two very different machines. In 2008, they used it for the Aluminum MacBook (the first Unibody MacBook), and then they recycled the same number in 2009-2015 for the 13" Unibody MacBook Pro.
The two versions have nothing in common except the Apple name, aluminum cases and 13" displays; they have no common components between them.
- 2011-2012 13" MacBook Pro
As you can see, there are no overlapping part numbers between the two computers, because no matter how similar the parts look, they are not identical.
I too have both a 2008 Aluminum MacBook and an Early 2011 13" MacBook Pro, both with their own quirks, which might be solved by swapping parts between the two...but the parts can't be swapped. I feel your pain.