A quick and hopefully final update on my experience with this since I made another large change a month ago.
I had grown concerned about the inverter transformer core saturation issue discussed above. The inverter was running hot, and I think the dirty AC power was stressing some items in my house, like the 'fridge compressor motor.
So, when I saw a good price on a 48 volt, rebuilt 120 volt AC inverter, I bought it, and ran an AC wire from it to the wh.
The obvious drawback of this is that it increased the cost of the project from a few tens of dollars to a few hundred. But there were some advantages. First, the voltage fluctuations and pulsing issues on the main inverter were solved, unsurprisingly. Second, because the new inverter runs at 120v, the load was smaller (1000 watts into a 4 KW 240 volt element). This turned out to be an advantage because the smaller inverter goes into steady, full-power mode with no pulsing and steady voltage more quickly, as long as sun conditions are good. Finally, my main inverter has less to deal with and more available capacity.
One thing I had to change was to put the offset in the Classic AUX settings up to 3.0 volts. At lower offset settings, the element wasn't getting enough power; basically if it's not getting the full 120 volts for much of the day, it won't make a significant contribution to the water heating. Setting the offset wide pretty much insures that it's receiving full voltage when the surplus power is available. Setting it too narrow did not do this, even when surplus was available, I think because my charge controllers could not respond quickly enough to the pwm-induced changes in load to keep the voltage from fluctuating.
In retrospect, I might have gotten a similar result by wiring the element at 120 volts just using my main, 240 v. inverter. I never tried this, but that would have lowered the load and might have kept it in a steady full-on state more often as well. But I don't mind that I did it this way; still plenty of hot water and now, plenty of spare capacity and no stress on the rest of my equipment.