Easiest probably would be to remove the existing plug (if any) and wire to the battery. Yes, you need a fuse a close to the battery as reasonably practical to protect against fire in case of a short in the wiring to the inverter. (The inverter presumably has a DC fuse or fuses to protect it from catching fire in the event of an overload or fault.)
Any connector rated for at least 50A would be suitable to use, but you'd still need wiring to connect between the one connector and the battery, and a fuse to protect that wiring. Using standard AC electrical connectors is not recommended due to the possibility of someone mixing things up accidentally at some point. One example of a fairly readily available connector that would be entirely suitable is the SMH SY50.
A 12V lighter plug is limited to somewhere in the vicinity of 10A or so, and would be ill suited for running the inverter at full rated power. For a low draw, such as many TVs, it should be usable but not ideal; they tend to be a bit unreliable at times.