Lifting Bag Size
Average hot air balloon needs 64 ft^3 to lift 1 pound with an internal temp of 100 degrees F.
400 tonnes = 881,000 lbs -> 881,000 lbs * 64 ft^3/lb = 56,384,000 ft^3
Assuming your lifting bag is 3x the length of your hull...
3 * 105 ft = 315 ft length
... and assuming a cylindrical lifting bag (sharp edges, unrealistic), cross sectional area of lifting bag is...
56.384 * 10^6 ft^3 / 315 ft = 179,000 ft^2
... diameter of lifting bag is therefore...
2 * ( 179,000 ft^2 / 3.146 ) ^ 0.5 = 477 ft diameter
... which is a wide load, might want to make it longer.
Fuel use *This is going to be quick and dirty on the heat transfer, in reality heat loss would not scale linearly and fuel consumption would be lower.
Average hot air balloon (77,000 ft^3) consumes thirty gallons of liquid propane an hour.
56,384,000 ft^3 / 77,000 ft^3 = 732 average hot air balloons
732 * 30 gal = 22,000 gal/hr
... with propane you get 91,330 Btu/gal...
... with hydrogen you get 51,585 Btu/lb...
91 Btu/gal / 51 Btu/lb * 22,000 gal/hr = 39,254 lb/hr or ~18 metric tons an hour of hydrogen burned to heat your bag.
18 tons/hr * 16 hr = 288 tons of hydrogen for sixteen hours of flight time
Your aircraft is only 72% fuel by mass, good enough for government work.
Heating Element Viability
... 72% mass taken up by fuel is unfortunately likely not good enough, even for government work. Your heating element is viable--heat is generally pretty easy to obtain from an energy source, especially when heating a large volume like a hot air balloon--but your understanding of using a battery to recover hydrogen from clouds, then using that hydrogen as a heat source needs work. Implemented as is will lead the crew of your airship to a fatal end. All the energy in the system you've described will need to either be A. contained within your hydrogen storage or B. contained within your battery at the beginning of your flight. It would be a better idea to just carry more hydrogen instead of the battery weight and associated machinery for performing electrolysis.
As a further suggestion, I would strongly advise you to investigate replacing hot air as a lifting gas with something that doesn't require constant fuel input. Helium will do, so will hydrogen-- 64,000 ft^3 of helium will lift 4454 lbs., while the same volume of hydrogen will lift 4808 lbs., both nearly four times as much with no fuel consumption to boot. That reality is why (unheated) hydrogen and helium are the lifting gases used in real world airships; I have not seen a hot air balloon of similar size, though I have read about theoretical floating cities in geodesic domes held aloft by a slightly higher internal air temperatures (which are neat). I also agree that airships are awesome; read some thermodynamics textbooks, learn like you'll live forever, and fly safe.