We have the same batteries, generator, and converter. Yes, the LP-powered generator will charge all batteries, but the coach battery pair has priority. The generator feeds the 120v system, same as shore power does. The converter takes the 120v and puts out a 12v DC charging current. The converter is wired directly to the coach batteries under the hood. When the coach battery voltage is over 13.4 for a minute or so, the battery separator under the hood will connect the coach and starting batteries together.
So if your coach batteries are run down, their voltage will not get to 13.4 for several minutes. During that time the separator remains disengaged and the starting battery will not get any charge until coach batteries get mostly recharged and the voltage gets over 13.4. Usually takes only a few minutes.
Want to know if your starting battery is really getting charged? Stick one of these in the cigarette lighter outlet in the center console: INNOVA 3721 Battery and Charging System Monitor, about $20.