Does the milky way have relativistic mass against galaxies which are moving away from it at high speeds?
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.