Will bootloader communicate via TX0 / RX0
Problem solved. The standalone CH340-based converter for whatever reason isn't man-enough to reliably pull down the RX0 line when connected to the Mega 2560 (also with a CH340 converter on board). I've seen suggestions posted in various places that this can be a problem with respect to DTR (though I don't have a problem with it). My solution is to insert a 7417 open-collector non-inverting buffer between the output of the stand-alone converter and the RX0 input on the 2560 board. I've not been able to find a schematic for the Mega 2560 board to determine its pull-up resistor size but imagine it must be relatively small. Thanks to those who made suggestions.