Hello: You’ve got some interesting thinking going on 😎 Looking at the diagram, are you pumping from left to right? This would be using the heated new tank to heat the old tank, but not recirculate around the house. Also, this way check valves would prevent bypassing when water is drawn. I’m not sure you need two check valves, but isolation valves in the system would be good so either tank could be off line. As to pump control; why not use the lower thermostat if the pump is there only for tank heating? This will keep the whole tank hot. The pump should only run to make up for standby loss, which will be little if everything is well insulated. I see no reason why the thermostat wouldn’t take 120 VAC, but do protect terminals from shorting. I’d avoid options one and two. They will waste energy. If the new tank has a magnesium anode, you can leave it. I’m no fan of aluminum. I would add “too much” insulation to tanks and piping. You’ll have a lot of surface to lose heat from and need to deal with that.
Now, what can be done to make the plumbing system more efficient? Why are you running low on hot water? Do the pipes need insulation? Can you fit in a shower heat exchanger or do cold water washing? Answers to that sort of question might save the need to do all this interesting work 😉