I very much doubt that a "cup holder" sized inverter can supply the rated loads that come with the product (200-400w). I've used a variety of small inverters and have never seen one of small size that is rated to that amount of AC power.
Even with the stated loads you are using, long lengths of DC wire may have too much voltage drop to operate the inverter. As others have said, locate the inverter close to the batteries and use large wire. How did you terminate the cig outlet to the 12v cable? What devices are also powered by that wiring?
Typical cigar lighter maximum amperage ratings are in the neighborhood of 6-10 amps. That is 70-120 watts at maximum and, with contact losses may be under the draw of the inverter. Don't use a cigar lighter receptacle, if possible.
Smaller pure sine inverters don't cost that much more than a comparable MSW inverter. Electronics will like a pure sine inverter much more. This is one choice where a bit more money is well spent. Don't buy the cheapest you can find, get a known good brand like Gopower, Samlex, Xantrex, Tripplite, etc.
Good luck with your 12v power.