A random idea: give the Eagle an extra maybe 600 flux dissipation - and a built-in hullmod that doubles the DPS of beam weapons, but also triples their flux costs.
Since the Eagle can already carry 1000-range weapons thanks to its ballistic slots, we don't have to worry so much about the long range of beams being oppressive; the Eagle can afford to buff its beam weaponry without becoming overpowered, and the flux cost increase should be roughly balanced with the extra dissipation. While at the same time, that extra dissipation also makes it more reasonable to use pulse lasers or even a heavy blaster or two.
I'm not sure how much this would help, but it is at least something that should be easy to try out...
Like the guy said... people would probably just mount heavy blasters instead.
I don't think beams need to be buffed in this way. Maybe they need a buff in general, but not for this specific ship - not when there are more straightforward alternatives.
The Eagle and Falcon are just vehicles for HVD's. That's what they're good for. They put hard flux on far-away targets and flit away from return fire. The only difference between the Eagle and the Falcon is that the Falcon actually succeeds at this task because the Eagle is too slow and can't tank to compensate.
Random riff on Ion Beam/Pulsers because apparently some people hate them...
They exist for a very specific reason: They make life suck for ships that drop their shields. That's why Ion Beams synergize well with HVD's, especially when you're supporting another ship that's doing the actual frontline fighting.
That's also why I put ion pulsers on my Doom cruisers: so that when the enemy's flux is too high to maintain a shield, the Doom can drop reapers at point-blank range with a little extra assurance that they won't lose as much armor from return fire because the other guy's weapons or engines are sparking with EMP. They're also not bad for neutralizing fighters for the precious few seconds it takes to go back into phase. The value from the pulsers isn't in dealing out damage - it's in adding combat longevity to a ship that doesn't have regenerating shields and needs to conserve armor.
Gravitons are the only other medium beams that make sense, but I don't like their style. In a simulated dogfight, they can appear to buy you an extra few seconds in a long fight by increasing the other guy's hard flux... but, for my money, the tradeoff of lower soft flux damage is worth it for the EMP effects.