You're asking a lot from a single lighter socket. I'm pretty sure the socket would be fuse-protected, so you shouldn't melt the wires. But you may "go dark" all of a sudden; and as you know, this always happens at the most inconvenient time.

For ongoing use, I think the safest and most reliable approach would be to have your mechanic add extra fused circuits right from the fuse block. This is not at all complicated, so if you're handy and can test for polarity, you could do it yourself. And if your big inverter is the type with a DC power cord, you might even consider wiring it into the fuse block and hard-mounting it in a convenient place.