I see. I used an electronics calculator that I found in some forum somewhere (similar but less awesome than eCalc) and it led me to those motors. I'll definitely use eCalc this next go around.
None of the calculators are perfect, but my impression is that eCalc is the best. For a few dollars (30 day subscription), I think it's worth it. (Note: I'm not associated with eCalc. Just a happy user.)
Lets say I was to redesign the power system... In what order would I choose components?
(props>motor>esc>battery)
I think the first thing to do is to sort of define your copter's mission. An acrobatic mini quad is going to have a different mission than a hobby flyer. A quad that is designed to carry a big DSLR on a gimbal system is going to have a different mission still. The main parameters might be things like flight time, stability vs. acro, cargo capacity, and all-up-weight (AUW). In your case, I think you would be in the low-mid range for cargo and mid range for AUW. It seems like you would prioritize flight time over speed/agility.
If flight time is your priority, then you need to start focusing on low kv and large props. I added up your gear, minus your power system, and came up with 850 g, if memory serves. So put into eCalc a weight of 850 g, "not including power system", and then play with different power systems to try to come up with longer flight times. Make sure that your hover throttle is around 50%. eCalc will not complain unless your copter would require more than 80% throttle on low battery, but I think most of us would find that to be unacceptable.
When I designed my quad (haven't built it yet, mind you), I wanted a flight time of around 20-25 minutes. In the images below, you can see what I came up with in eCalc, and the resulting flight times, etc... Notice a couple things. First, I am using 14" props with 390 kv motors. Second, I had to step up to a 4S battery to push the motors hard enough to keep the hover at around 50% throttle. Third, my airframe, minus power system, is only around 450 g. This is for a home-built H-quad frame, made out of wood. I haven't built it, but I based it off of other people's builds. My AUW comes in around 1.5 kg, with a hover time of 24 minutes, and a mixed flight time of 13 minutes.
I'm open to a complete redesign if I will get the flight times that I want.
Take this with a grain of salt, as I am just getting started with quad design, but I think your frame is holding you back here. That frame is heavy for its size--you should be able to come in under 400 g, I think. But the main way that frame hurts you, IMO, is that it limits you to 12" props. I think you are going to be challenged to get the kind of flight times you want using 12" props. If you were to cut a lot of weight and go to 4S battery, maybe. But given that switching frames may be one of the main ways you can cut weight, you may as well consider whether you want to go with larger props as long as you're at it. Or maybe you can tweak things out in eCalc to get something that works with 12" props, by using different motors and ESCs. It kind of all depends on what exactly you consider to be a long enough flight time. 7-9 minutes is plenty for a lot of people, and they just land and stick another battery in. If you are intending to do long FPV flights, like 15-20 minutes, I think you are going to have to really go back to the drawing board. That is an exceptional flight time for a multirotor, and they really have to be purpose build to achieve that.