I looked up the top 10 results for 400w inverters in Google. Check the specs. 400W is continuous. Most will surge to 600-800w, and we have no idea what she is going to connect to this inverter. It could be something minor or it could be a continuous max draw with random spikes. My 800W will hit 90A for a half a second when tossing the switch, then will settle down, and that's with nothing connected to it. I have a shunt on that circuit. Yes, my inverter has dual fuses 100A from the factory, one on the feed, on on the earth. Neither have ever blow, but when I dropped a butter knife and it landed just right on the small exposed part of the lugs not covered with the nuts, it blew my fuse at the battery instead of welding it across the terminals and melting down my rig.
It isn't any different than a compressor fridge that hits 2-3x the draw for a few seconds when it kicks on. If you don't mind replacing the fuse every couple of times, then go for it. I'd rather not. Size the wire to be safe for the max draw of the unit, then size the fuse a little smaller to protect the wire, as close to the battery as reasonable. I'm not saying run 00g and a massive 300A fuse, but staying within the manufacturer's parameters for the device. Decide what you feel is the acceptable risk.