Lets introduce some real world facts and physics...
You clearly stated and I quote
The handheld version of "my design" with even the cheap parts I'm using would still only weight about 10-15lbs at worst, if I had top of the line mil spec caps, bats... it would even be smaller, lighter and more powerful.
This is going to be a necessity if you goal is to replace traditional infantry weapons... Again not an option a requirement to keep weight to a minimum...
Lets forget about your exaggeration of mil-spec items for a minute as once again physics set the rules not the military... Yes, the military can trim a few corners over mass produced consumer stuff (dependent upon cost) on some things but they can't defy physics and shrink 10 lbs of poop into a 1 lb block... There is no huge imaginary gap on the battery and capacitor tech as you are portraying...
Water weighs about 8lbs a gallon... How much do you plant to use to cool this? Remember less water = less cooling potential, and even at 1/4 of a gallon you just consumed a large percentage of your target weight and have very little cooling ability overall... If you are to abandon the water cooling in the final design, how much heat sink material do you plan to use? How 'bulky' is this going to be?
Copper isn't exactly light weight, to give you a real world idea, 150 pennies (3 rolls) weighs about a pound... How small/big are your coils? How many pounds of copper are you using in said coils? Your 120 foot roll of 14g wire is about 1.5 lbs in it's own...
Battery, you stated
these are small 4cell lithium batteries called ballistic evo2, I'm gonna do 3 in series so 40v 125amp.
Those batteries weigh in at about 1 lb a pop... There just went 3 lbs or about 1/4 of your target weight...
How about the rest of the guns construction? The barrel, maybe just a few pounds there? Care to elaborate on what you plan to build it with that can handle the forces it will experience? Remember a bulk of the weight of a traditional gun is simply the 'frame' something you will need to nearly replicate if forces are to be equal, but you also now have to integrate the entire propulsion system into said design and maintain the same weight, how?
How long is this gun going to be? How many stages? I believe (memory recall) earlier we were at about 16 feet of barrel length and 100 stages, you need to get this down to the 2 foot range for a hand held weapon... 100 stages in 2 feet means each coil has to be about 1/4" long, not much room for strong producing coils in that limited space...
Again you really need to step out of imaginary land and return to Earth and our reality where physics apply...
Bob, explained a huge real world issue (as have others) and you just gloss over them as if they don't exist and steam roll forward as if the problem was solved just because you ignored it...
This is tin foil hat land mentality, not reality...
Let the helium out and return to Earth, build a working prototype that is superior to a 22 cal pellet gun before you brag about it being a 50 cal replacement or defense system for a naval fleet...