Yes, you are probably correct.
When starting the car, the auxiliary battery gets boosted quickly from the traction battery,(allowing the system to run) but the handshake process to make sure the charging is safe (waterpump,fan, (AC?)) might draw too much out of a weak 12V battery. Could even be a contributing factor in damaging the charger(relays not closing firmly or chattering on /off). Just speculation of course,but good reason to have a sound auxiliary.