The Champion, with a flux distributor, has 940 flux. If it has 2 HAC and a Plasma Cannon its got 428 Flux from the HAC and 825 flux from the Plasma Cannon. This nets a long term total weighted shield DPS of 1204. Plus 512 for the squall while it lasts. Makes 1716. This assumes its shield is not raised. You get a bit more out of HEF but not enough to quibble about an extra 7.5% DPS from the Plasma Cannon. Up to 1247 consistent DPS and 1761 when the squalls last.
You should check your numbers before accusing me of 'not typing them out because I am wrong' (or look back at the number I typed out previously that you apparently didn't bother to read). 2x HAC + plasma is (214*2)*2 + 750*1.075 = 1662 DPS to shields, compared to 3x HAC +2x grav + 1x Ion: 3*(2*214) + 2*(2*100) + 50 = 1734 (only 70 more shield DPS with 450 of that being soft flux which is significant). Given the soft flux, I would call that about the same. With squalls, champion has 2174 shield DPS which is a lot more. Also, HEF is burst DPS, not a flat small bonus, which also does matter a lot for winning the flux battle.
Not to mention that champion build is not specialized for shield DPS at all. If I wanted to hyper specialize into shield DPS like your build, I could run volatile particle driver for 450 kinetic DPS at 450 flux/sec which would give 1823 sustained shield DPS for 876 flux/sec (and very high burst too). Or if I don't have one, I could run autopulse and 4x IR pulse lasers for 10 more OP to get 1823 sustained DPS for 1158 flux/sec and 2335 DPS with squalls (before you bring up range I would probably do something like IPDAI + elite PD to match ranges with IR pulse lasers). Those would both be more fair comparisons to your build, but having plasma cannon for real hull/armor DPS is better overall IMO.
Then you can also consider burst DPS, which is very important for the flux battle. The autopulse loadout has 3113.5 upfront burst DPS, and if you consider HEF burst (50% damage) with autopulse burst, it's 214*2*2 + (150*4 + 1500)*1.5 = 4006.5 burst DPS to shield, before even considering squalls. Of course, that's ~2158 flux/sec too, but it's trading at 1.86 damage/flux which is very efficient and will win the flux battle unless you are shooting into fortress shields.
Like. An AI Champion fit as you describe cannot kill an AI Eagle in the sim before it runs out of squalls when the AI eagle is fit with only Heavy Mortars in the front and gravitons in the back which is an absurdly bad fit and i think that the only reason it will kill it before time runs out is because the Eagle is likely to get wedged on the side.
You can just say 'the eagle is fast enough to run away the whole time', which is what happens, and was already established by the 'eagle is faster' statement. Not sure what you are trying to say with this? Eagle is better at harassing/running away? Sure, but that's not what we were talking about.
Also, if you want to talk about skills, I put an officer with ordinance expertise into my ship and get the skill that give bonus dissipation as well. I run this champion build and flux is not a problem with skills. And of course you can do that on an eagle, but what are you going to do with that extra dissipation from skills? The medium energy weapon options are all short range and not very good for eagle. I would rather have a plasma cannon to utilize that dissipation than some pulse lasers or phase lances. I can also get missile weapons specialization to double squalls (and increase damage and rate of fire with elite spec), and systems expertise to increase HEF uptime which increases sustained bonus damage to 15%, and the extra charge also means the initial burst DPS of 50% spamming charges lasts 15 seconds (3 seconds with 4 charges and you get one back). Plus energy weapon mastery to increase damage even more, and elite to reduce flux cost further if that is still an issue. So champion benefits a lot from skills too in ways that eagle does not (at least nowhere near as much). Definitely not fair to just consider one skill that benefits eagle more and ignore the ones that benefit champion more.
Range is definitely a significant factor, but so is burst DPS, and so is hull/armor DPS that you are conveniently choosing to ignore in your comparison. I suppose the comparisons is fairly subjective because so many categorically different factors need to be weighed against one another, but I have personally not found eagle to be very good, while I have found the champion to be very good.
The only way I see eagle being better is in harassing and chasing down small stuff due to the speed advantage, but I think it's still not really that good at that, and there are better ways to achieve that IMO.