Hi, Read the manual for your genny. While some of the smaller generators do have 12v out, some are labeled "Not for charging batteries". My guess is the 12v is 12v and not 14v, so the battery will not get charged.
As suggested, use the converter in your TT to do the charging.
Used to have 2011 189FBS &
2011 Taco 4WD Access Cab.