Most electric water heaters don't use digital controls. I'm not aware of a remote control for a standard electric water heater. Here is what I would recommend: Install a water heater blanket to increase the insulation on the tank and turn it up to 140. I would only do this if you don't have small children or elderly in the house. The potential for scalding is serious. You could also install a mixing valve near the heater to prevent scalding problems. Also, make sure that all of the pipes are insulated. You don't want to be losing the heat before it gets to the tub. Sorry to be use your situation to make a point, but this is a classic case of really bad plumbing design on multiple levels. My personal opinion is that 90 gal soaking tubs and multiple shower heads are hugely wasteful to the point of gluttony. That said, if a builder or homeowner is going to install them, the plumbing system needs to be designed for them. Water heaters in the attic are an invitation for damage from leaks and difficult to service or replace. Plumbing and heaters in unconditioned spaces invite freeze damage and energy waste.

Tankless heaters have used as a marketing idea that they will increase efficiency by not having hot water sitting around in a tank all day not being used. It's an easy concept for consumers to grasp, but it's not true. On an electric heater, the difference in the energy factor is very small between a tank and a tankless heater. A tank heater will be in the area of 0.93 and a tankless might be 0.95 - not enough difference to talk about and certainly not enough to provide payback. That represents the amount of heat lost through the walls of a well insulated tank. With gas heaters it's different. The vast majority of the efficiency gain is in the lowered flue temps. Second is not having the heat loss through the center of the tank. For your situation, what you should know is that a properly installed and operating electric tank heater is extremely reliable and efficient. Get yours replaced or repaired and the TPV properly routed to the drain so that it doesn't cause damage. Since you have lost confidence in your heater, I would just replace it rather than repair. Make sure that you are addressing thermal expansion issues that might have caused the TPV to open. If you want more space, consider a stacking washer and dryer and eliminating the sink. The front loading LG's are full size (huge inside) stack and don't use much water - eliminating the overflowing washtub issue. That would be a cheaper way to get space than a tankless heater and they are great machines.

Here's one where different public health recommendations for different purposes are inconsistent. The 120 deg recommendation is to avoid burns to small children. But from what I've read, that's too low to kill bacteria. I'd hate to think of what's growing in a tank that's stagnant because of infrequent use. How often are you there, how much time passes in between visits? If it were mine and hadn't been used in awhile, I'd probably flush out the tank by running the hot tap (which is cold) following arrival, then turn it up to a good med high setting to cook whatever is inside, and then turn it down after a few hours so that it remains at a safer temperature.

From what you write, you may know this already, but an electric WH is wired so that the top thermostat is the one that takes the lead, heating the top half of the tank for a quick recovery. Any time the top water is below its setpoint, the top element is on and the bottom one is not. The top thermostat is actually a double-throw switch. Once the top thermostat is satisfied, you have half a tank of hot water. The top thermo reroutes the incoming power from the top element to the bottom thermostat. The bottom thermostat then hammers the bottom element until it's satisfied, then it shuts off. Now you have a full tank of hot water. This allows you to have 2 high power (4500 Watt or whatever) elements but a WH that never uses more than 4500 Watts. It sounds like your tests seemed to show the elements and thermostats working right. Now you need to actually monitor what the upper and lower elements are doing while the tank is in operation. This requires working with hazardous voltages, so while I don't want to offend you by suggesting anything, I have to say right here that if you don't know electricity and electrical safety by heart, you should turn this over to a qualified service technician. That warning out of the way, this is pretty easy to test with the access panels off,; insulation peeled back; and neon test light, DMM, or Wiggy at the ready. For something like this I'd probably connect a test light to each element with insulated clip leads, so I could watch the transition from top to bottom element happen (or not happen). My guess: you'll find that the top thermostat's contact that closes on temp rise, the one which transfers power to the bottom element, is not closing properly or consistently., Another possibility is that although the bottom element measures the right resistance for 4500-5000 Watt WH, and the top thermo is working right, the wiring between top and bottom elements has failed. Probably close to 50 years ago, I rewired a WH on which that had happened. Maybe the wire that failed had been nicked when it was pulled in at the factory or something, but for whatever reason, it had burned in two about 1/3 of the way from top port to bottom port. Let us know how you make out with this.