Do you plan on running the inverter at full load, even if for a short time? That's a heck of a lot of current.
3000 / 12 = 250A. We have to add in some degree of inefficiency, let's say it's 90% efficient being a high frequency type (most likely), now we have 278A. Next add in some margin as dictated by most good advice, we now have about 420A. This assumes a fuse over rate of 1.5, some people may have differing views, 1.25 (350A) for example, so consider all of them. Ambient temperature derates a fuse so ensure it has good ventilation too.
You may find a heavier fuse is more appropriate to guard against the brief very high current the inverter will draw if it is connected to 12V directly without the capacitors in the inverter being pre-charged. There's a lengthy thread on the forum discussing ways to pre-charge inverter capacitors.
I'm building a mobile-ish setup that will have a battery disconnect (one of the big RV / Boat from Blue Sea or similar) switch between the battery and the inverter charger. I'll leave it off most of the time. I would like to avoid the high current inrush and spark at the switch when switching...
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The fuse is there as a last barrier in case things go horribly wrong. If you have a very large fuse your cabling needs to be able to cope with the currents up to the point where the fuse goes away, indefinitely. It must also be heavy enough gauge to be able to exceed the fuse rating sufficiently, in the short term at least, in order to actually be able to blow the fuse.
At full tilt this inverter will seriously reduce the amp hours available from a 300AH lead acid battery and need a quality/beefy lithium bank too.
Here's a guide for you to read.