This solution assumes the following
a-The scaling factor for the object is correct i.e real world dimensions .
b- There is a .STL file available .
c- This will take some 3ds max skill and time .
Steps to be followed -
1- Take the .stl file into Autodesk 3ds max .
2- Now we are going to trace the exposed surface using a process called reptology . Here is link to get you started - https://www.youtube.com/watch?v=azkbuZloTqo .
3- Once you are satisfied with recreating the mesh and sure it is as accurate as humanly possible , run the measure utility on the recreated part .
This is the link . https://knowledge.autodesk.com/support/3ds-max/learn-explore/caas/CloudHelp/cloudhelp/2016/ENU/3DSMax/files/GUID-12CDEA7A-042E-4CB3-9BF6-8191D3290CD8-htm.html .
If there is a upper side and a lower side to the model then you should divide the surface area in half ( in this case it would be ).
3Ds max is very accurate if your part is designed properly .
4- This can be done directly on the .STL file if you can efficiently isolate your target area , lemme warn you this is not an easy task .STL files are notoriously difficult to edit. ( its much easier to redesign on top )
Try it should work out fine .
Please do not waste time processing .STL files on 3D-Slicer, Seg3D, ITK-Snap
they are mostly for file preparation i.e (.STL file generation ),viewing and study purposes only .