The wire gauge may be causing a large voltage drop, large enough to cause the inverter to not start up.
for 120W bulb, you need at least 10A at 100% efficiency, so assume you are drawing 12A or so from the battery. A cigar lighter socket is only rated for about 10A, but close. However, those may have poor contacts and cause a bit of voltage drop as well.
Even with 14G wire, you get a voltage drop of 0.75V at 12A in each direction (1.5V total). If using smaller wire, the voltage drop will be even more. For example, 16G will drop almost 1V over 10ft (each way), so for both wires you will get close to a 2V drop.
To have reliable inverter input voltage, use as large a wire as you can, 12 or 10 gauge would be better.
Another point: Some inverters are modified square wave, not true sine wave. Some chargers will work with square wave, but poorly or not at all. If the charger works with the inverter plugged direct to the battery, then the charger is probably ok with your inverter. It is just the inverter then that does not like the long wire length.