no need to worry. The amperage going through that wire to your auxiliary battery will be limited by the voltage that your car allows your alternator to put out. In other words, the voltage at which your alternator is at is controlled by the voltage regulator on your alternator. Therefore, if the battery your voltage regulator is connected to, e.g. probably your main car battery, is not really that depleted, then your alternator is not going to put out that many volts. Consequently, the amperage going over that wire from your car battery/alternator to the auxiliary battery on the trailer is going to be fairly minimal. This is compounded by the voltage drop caused by such thin wire.
I had this problem when trying to charge the battery on my travel trailer. The voltage (as detected on my Scan Gauge) that my alternator is putting out is only briefly 14.0+. As soon as the car battery is refilled within a few minutes, the voltage drops fairly quickly--to about 13.4-13.5. Factor in the voltage drop caused by the thin 12-14 gauge wire on my 7 pin connector, the voltage at the trailer battery was very low- resulting in very little amperage going through. I ended up getting three 10 gauge wires to reduce the voltage drop.
In sum, from the perspective of too much amperage, I wouldn't worry about the wiring melting or anthing. But you may want to upgrade the wiring if you need the trailer battery to charge quicker.
2007 Toyota Tacoma TRD 4x4 Dcab