There are a number of ways to design such a beast. It's an iterative process, and it depends on your constraints.
For example, you could choose to constrain your motor selection to a given motor, which will then dictate the optimal cell count for the optimal RPM with the optimal propellor for the optimal thrust. Once you determine the thrust with that setup, you can determine the all up weight of the system (assuming an airframe weight and electronics weight as you have done), and see how much thrust remains for lifting your payload. I think you'd want to design the system so that it had a 30-50% margin available for lift beyond the weight of the airframe and the payload.
Let's say the aircraft really does weigh 60lbs. If we say it has a lift target of 200 watts/lb, that means it's going to consume 60x200 = 12,000 watts. At 12S, that's 270A total. The load is spread to 4 motors, and 4 ESCs, which is only 67.5A per motor/ESC. 12,000 watts spread over 4 motors is 3000 watts per motor.
If we agree that 200 watts/lb is as much as you need, then you can actually use smaller motors, and smaller batteries, and smaller ESC's to keep the weight of the system down. This will reduce the power consumption of the system. So you can repeat the exercise by figuring out the weight of the system with smaller motors.
By your calculations, you have 15lbs of battery. If we assume ~700g for the 6S 5000 battery, you seem to looking at carrying about 10 batteries, so a 12S 5P setup.
The 12S 5P 25,000 mAH setup can provide 20,000 useful mAH capacity. If you consume 12,000 watts, this battery system can provide power for
20,000mAH / (270A * 1000mAH/AH) * 60min/H for 4.44 minutes.
If you assume that it's really 270 watts/lb, then you're talking about consuming 16,200 watts, 365A, and 3.28 minutes flight time.
It is true that you'd need to find pusher propellors to match. So make sure you can get them (or make them if you must) before you get too deep into it.
I had thought about doing this sort of large multirotorcraft a different way though. I think that using 2 bladed flybarless rotor heads would be excellent for this purpose.
I fly some relatively large helis, like my 40lb Air Crane, which has a 6 bladed head, 2000mm disc. It uses a NEU2230 2Y motor, which is capable of 10,000 watts. But, I actually only consume 1950 watts flying around. In general, nearly all of my scale helis consume ~50 watts per pound. My sense is that you could use rotor heads, and budget 100 watts per pound, and have a craft that works beautifully.
There is more mechanical complexity if you choose the rotor head route. You need servos to change the collective pitch of the blades. If you make the servos all move the same direction, then you don't need any special electronics to handle the collective pitch, because no mixing would be required. You'd simply have all servos for a given head connected via a Y harness, and "throttle" would control blade pitch. And real throttle (RPM) would be governed by the ESC and set constant.
At any rate .. I think it's quite doable. And perhaps, you can make the airframe / motors smaller than you think, which will use less power and require fewer and lighter batteries. It all helps.