Does the milky way have relativistic mass against galaxies which are moving away from it at high speeds? According to the article cited behind, and to a post here in Astronomy SE , there are galaxies moving away from Milky Way faster than light, even at speeds of 2.3c . According to this article 
Can two galaxies move away from each other faster than light


  The two galaxies we've been discussing are not travelling through
  space, it is the space between them that is expanding. Or put in
  another way, they are stationary and all the space around them is
  being stretched out. This is why it doesn't violate the theory of
  relativity, because it is not motion in the traditional sense.


So does this motion between galaxies "in not a traditional sense" count at all to calculate relativistic mass?
What the mass increase from special relativity tells you is that if you see a physical system (say, a rocket with a physics laboratory on board for convenience) flying past your location at a certain relative velocity and you measure the mass of an object in that laboratory, for instance by measuring its gravitational effect on another mass in the same laboratory (using lasers to measure distances and velocities and so on), you will get an answer which is higher than the one an observer in that laboratory would get.   

In and of itself this doesn't even tell you much about what you'd measure about a far distant laboratory, especially one far enough away that the expansion of the universe significantly effects your measurements. 

Furthermore, since these two galaxies on opposite sides of the observable universe cannot actually  exchange messages and observe one another, the mass increase is not really a consideration -- you can't measure the mass of an object in the other laboratory, because you can't get a signal there and back.