Ted, looks like you have several issues. With a new alternator you should be pumping out lots of current to the battery when the engine is at or above fast idle. Headlights on at idle is tough on a battery that is already noticeably depleted but as soon as the rpm's come up, the ammeter should show a net charge going to the batteries. If this is not happening you have either 1) a bad alternator/voltage regulator (even if it is new), 2) bad batteries, 3) bad cables, 4) a bad diode or switching unit that controls charging of the chassis batteries, or 5) a bad ground. Are the house batteries recharging while you are driving? If so, then the alternator/voltage regulator are probably OK and I would start focusing on the other areas.
If your electric heater is like mine, and it may not be, it is a marine electric heater and the electric element could be burned out. Replacing the element is somewhat of a pain but not nearly as bad as replacing the tank. Be sure and check the breaker to see that it isn't tripped and measure the voltage at the tank from the incoming 120 volt line to be sure you don't have a large voltage drop.
The use of an electric water heater on 20 amps is a problem since about 50% of you available power is being used by the water heater when it is on. A tankless propane water heater isn't a bad idea BUT the one you are looking at is likely not sufficient by itself to meet your hot water needs. I base this comment on the fact that they need a minimum of 55 degree water coming in (which you won't have during the winter) and that the output is only 5 liters per minute with a 25 degree C temperature rise. That is a pretty low flow rate meaning pretty pitiful showers (either low flow or low temperature).
There are more powerful tankless propane water heaters that you might check out. You will need to have those vented to the outside though. I would not want an unvented propane heater in a bay. You might take a look at something like the
Takagi tankless water heater that is rated at 3 gpm with a 77 degree F temperature rise. A unit that is made to fit in place of a 10 gallon propane water heater (which would not be the best configuration for you) is the Precision Temp RV-500.
This site has several different models to choose from. Storing hot water in your existing tank will not work with a tankless heater. The hot water that goes into the electric water heater will cool without power to the electic water heater and then you will have to overcome the 10-20 gallons of 'cool' or 'cold' hot water with the output from the tankless heater. You might be able to plumb the system the other way around and run the output from the existing water heater into the tankless heater so under 'normal' conditions the cold water just passes through the electric heater but if you need more hot water or the incoming water temperature was low, you could run the electric heater for 15 minutes or so to heat the water that would be coming into the tankless heater thus improving the temperature rise from the tankless heater.
Finally, your question regarding the Xantrex 3-stage charger/inverter is answered yes. It will replace the converter although if you have the space I would suggest leaving the existing converter/charger in place and just unplugging it. Without the converter, 12 volt needs will be met from the batteries and/or the charger portion of the Xantrex. I have been very pleased with the charger/inverter solution on my coach since I installed a 2kw ProSine inverter with a built-in 3-stage 105 amp charger.