[wiswind]

350W / 12V (worst case battery) = 29.2 amps.
The point of this calculation is that the rated current draw of the inverter is above the rating of the fuse protecting the circuit that it is on.
What this means is that the fuse will blow before any protection on the inverter will.
Normal voltage would be about 14V..... 350 / 14 =25 amps, which is still above the fuse rating.
It is NOT a problem that the inverter protection is above the fuse rating.....
I am only stating why the fuse is blowing.
The inverter should NOT be blowing a 15 amp fuse with what you are doing.

If the inverter is blowing the fuse with no load, then the inverter is defective.
I would try the inverter with NO load....then try it with a very small load.
With no load, or a small load, the fuse should not be blowing.
Do NOT put a larger current fuse in than the correct rated fuse !!!!!!!!!!

I would also try another item in the same cig lighter.....like a cell phone, etc.
If you still blow the fuse, then give the socket a very good look.

You could try the inverter in another vehicle....but you better have the correct fuse for that vehicle handy.