Honestly, if you want to up your firepower, you should start working on your fixed weapons skills.
Fixed weapons do a lot more damage. Gimbals are a "death by thousand cuts" type of deal - they are more consistent on hitting the target. Once you learn to aim fixed weapons to mitigate that disadvantage, you'll never go back.
If you want to stay with gimbals for now, bursts and MCs are about the best you're gonna get. (Beams are too hot for an Eagle)
If you have time, you can invest a couple of days of "work" down in the Imperial space to earn your wings with their navy and the Imperial Eagle variant, which has a Class 2 hardpoint up top. Having a C2 multicannon instead of C1 is a great upgrade.
If you decide to go fixed, whole new world of options unfolds. Things like rail guns, plasma accelerators and Frag cannons are on a whole new level of power and even a small ship like Eagle can cause a lot of trouble with them.