The Klingon Battle Cruiser is not weaker then the Federation Cruiser... thats a fact. So the only possible reason for you problem must be you. Instead of posting this QQ in the Klingon Faction Forum, you should go to the Battle Cruiser Forum and try to gather some information to improve your gameplay.
Nothing of the sort was mentioned in the beginning post, just that 1 cruiser couldn't take down another cruisers shields.... no levels were mentioned. But surely you would have noticed the large +20 on the target reticle??
Probably didn't have his weapon power very high. I was fighting a fed science ship yesterday during Borg Hunt and he was hitting me with spit wads. I realized though he had his shields at full 100-125 power because of his fast default shield recharge rate.
As you know, offense always wins that over time. At the end of the match the guy accused me of cheating, because he couldn't hurt me. I told him to check his power. He can cry all he wants, but I play a science ship on my Fed and knew exactly what I was seeing.
If you have no power in weapons, don't expect to do a lot of damage at Tier 5. The AI may roll over against weak firepower, but players won't. Lack of EPS also hurts those who use big cannons, better to stick with regular cannons and turrents than try to use the heavy stuff... you'll put out a lot more damage over time, particularly with buffs like rapid cannon II.