The biggest advantage to using the trailer's converter or a stand alone charger is the ability to "float" the charge as the battery tops off. I'm not sure, but, I would imagine the "12 volt charger" built into most generators is not as sophisticated as the converter or a good stand alone unit. I would think you would have to pay much closer attention to the gererator only charge.
Most of the RV friendly generators (either red or blue) are quite capable of powering the RV and charging the battery through the converter; I would fire up the generator and let the converter do the charging. One connection (the RV power cord), you don't have to have access directly to the battery terminals and minimal risk of over-charging (cooking) the battery.
My 2 cents, your mileage may vary...
2 Cats; J-Lo and Ragamuffin :R
2014 Thor Tuscany 40RX DP
2011 Ram 2500 Longhorn CTD HO
2011 Keystone Cougar 318SAB (now gone)
2008 FunFinder X 210WBS (Sadly gone)