Perhaps you could transform the surface integral to a volume integral using something like Gauss' law and then use NIntegrate, which works for an inequality constraint inside Boole.

No, it takes about 10 seconds before giving the DiscretizeRegion error. I'm running version 10.2. Could that be the point?

If it takes too long, it might work better to use NIntegrate with the MonteCarlo (or any of its daughters) method of integration.

Tomorrow I'll install 10.3 and try.

Thank you Frank. I think you might be right.

I rephrase your simple case as follows:

In[84]:= 
reg = ImplicitRegion[x^2 + y^2 + z^2 == 1, {x, y, z}]

    Out[84]= ImplicitRegion[x^2 + y^2 + z^2 == 1, {x, y, z}]

    In[85]:= NIntegrate[1, {x, y, z} \[Element] reg]

    Out[85]= 12.5664

And it seems to work ( $4\pi$ is the answer). But if I do the same with my integral (for instance for $a=0.5$):

reg = ImplicitRegion[Sqrt[1/3 ((x - y)^2/(x + y)^2 + (x - z)^2/(x + z)^2 + (y - z)^2/(y + z)^2)] == .5 && x >= 0 && y >= 0 && z >= 0, {x, y, z}]

and then

NIntegrate[Sqrt[
 12 ((x - y)^2/(x + y)^2 + (x - z)^2/(x + z)^2 + (y - z)^2/(y + 
      z)^2)]/(E^(x + y + z)
   Sqrt[(-((4 x (x - z))/(x + z)^3) - (4 y (y - z))/(y + z)^3)^2 + ((
     4 z (y - z))/(y + z)^3 - (4 x (x - y))/(x + y)^3)^2 + ((
     4 y (x - y))/(x + y)^3 + (4 z (x - z))/(x + z)^3)^2]), {x, y, 
   z} \[Element] reg]

Then I get the error: "DiscretizeRegion was unable to discretize the region \ ImplicitRegion[<<2>>]. "

Any further ideas?