You have another answer that advises not to buy from Amazon. Your description illustrates why. A Class T fuse is not a slow-blow fuse as stated in the amazon description it is a fast blow fuse. If you buy from Amazon at least try to find the manufacturer's specifications and check them. There is still some risk that you will get a counterfeit part.
If the manufacturer doesn't specify a fast blow fuse, don't buy one. They are more expensive than standard fuses. The fuse is not going to be fast enough to protect the inverter from damage due to an output fault. It is very difficult to protect power electronic circuits with fuses. The best designs have electronic protection. In some cases, the manufacturer will recommend a specific fuse, but that is not to protect the electronics circuit from damage. It is used to prevent the equipment from exploding or starting a fire if it fails. It is probably required by the UL listing.
The inverter manufacturer probably put a high minimum on the battery fuse because a standard fuse of that ratting can withstand the inverter's capacitor-charging surge current when it is first connected to the battery.
You should select it according to the battery manufacturer's recommendations. If they recommend less than 350 amps, get a different battery.
Added re fuse speed
It appears that most fuses that are not specified as "slow blow" may be considered to be "fast blow." The Class T JJN fuse is probably more expensive than you need because it is rated for a higher voltage than you need. Look at ANN and CNN fuses. They are rated 125 V AC / 80 V DC.