The heat indicates resistance. It means at that point, there is a chokepoint that the electrical draw is greater than the conductor it's being sent over can support. The tip is likely rounded and the very very tip of the rounded head is all that touches the socket. this creates resistance and results in excessive heat. It will also ultimately result in premature failure of your lighter socket and the power inverter plug end.
This is very common with high load 12v devices utilizing the cigarette lighter socket. correctly engineered ones have square or flat surfaced heads. There's not much you can really do. Maybe blob solder on to it and file it smooth?? Sadly, most junk made today is just that. junk. The now-to-market 3x80min charger HAS provisions for a ground but isn't chassis grounded! is it safe? abso-fing-lutely not. No UL approval, no nothing. Will people buy it up? Yup. Sure will.