Yes no surprise.
Sorry when I said the CTS I meant the DTR but when I said that didn't work that was on the CP2102, not the CP2103.
Let's assume for a moment that the CP2103 handles the DTR correctly.
The mistake you have made is to connect the DTR of the CP2103 directly to the reset pin. In fact it needs to be connected to the reset through a series capacitor, 0.1uF seems normal. This converts the logic level into a pulse and it is this pulse that resets the Arduino. With it connected directly your CP2103 is holding the Arduino permanently in reset.
The other thing that worries me is that the CP2103 is being powered by 3V3, which means all the signals out are at 3V3 logic levels which technically below the minimum high a 5V Arduino needs, but often you get away with that. However one thing you won't get away with is that the 5V TX signal from the Arduino is too high for the 3V3 maximum input of the CP2103's RX input. This needs to be cut down with a potential divider, before any damage is done. ( It might be too late for your chip ).
As you have a PCB of the circuit you must use the well known practice used in industrial development of altering the circuit using a sharp scalpel to cut tracks and thin wire to make any contacts.
You should be able to cut the track from the DTR output and bridge it with a hand soldered surface mount capacitor. A smiler technique should be use for the potential divider.