There are a couple of issues at play, Bob.
First, the internal Parallax converter isn't really a trickle charger, but it is a fixed voltage converter. At 13.6 volts, you'll likely get about 10 - 15 amps of charging current into a single battery. Adding a second battery will double the amount of current.
Converters that deliver a fast charge raise their output voltage to 14.1 volts during the bulk part of the charging cycle to push more current into the batteries, then drop back to 13.6 volts at the end to keep from boiling out the electrolyte.
Since the amount of current the battery will accept decreases as it gains a charge, you're better off recharging it to about 80% full. Gaining that last 20% of a charge will take forever. You'll be better off getting the battery up to full charge by plugging into electricity overnight when you get back from your trip. Or if you have a charging wire going from your car to the trailer it will top it off while you drive home. At 10-15 amps of current, you're looking at about 4 hours of running time to go from a 50% charge to 80% charge.
Adding a second battery will let you go longer before you'll have to recharge. You'll still have to run the generator 4-6 hours to recharge but you'll only have to do it half as often.
Putting a car charger across the battery may or may not decrease the amount of charging time. Most likely, the amount of current provided by the Parallax converter will decrease in direct proportion to the amount of current provided by the second charger, so the net effect is little or no gain in charging current. This is because the Parallax has good voltage regulation - it's voltage stays the same regardless of how much or little current it produces.
How far is it from your converter to the battery, and what size of wire connects them? Short, fat wires move more current into a battery than long, thin wires. If the distance between the converter and the battery is more than a few feet, doubling up on the wire size will greatly improve the charging rate.
You can also put an automotive ammeter in line between the converter and battery. This will not only let you see how much current is going into the battery while it's charging, it will also give you an idea of how much current you are using when you turn on a light, etc.