I too am having the same issue. I find that more times than not I can survive the first barrage and even get on a good run where I can take out a ship or 2, however my problem is lack of damage. I simply cant damage the ships fast enough once they start swapping taunt, pair that with Rex and JC's constant Protection restoration and its damn near impossible.
Currently I am farming Poe's Xwing for its massive damage to see if that helps get me over the hump but thats going to be a long farm. I am also working on Rex and Umbaran to see if they help.
My current ships are mostly 7 stars but admittedly my pilots range from Tier 6-10's.
https://swgoh.gg/u/refugeassassin/ships/?f=Light%20Side