When the power loss,the batt inverter not work

Some things to clarify:
Is the battery a 12volt or 24volt? I will assume a 12volt.
You should always have a fuse between the battery and the inverter. The fuse rating and wiring between the battery and your inverter depend upon the size of the load you will hook up to your inverter.
Inverters are usually NOT 100% efficient. Typically they are 80-90% efficient. If you run an inverter at rated load (your case 3000watts), you will most likely damage the inverter. If you run the 3000watt rated inverter at 500watts or less, you will have significant internal losses.
The capacity of the battery is important.
The type of battery is important (Flooded, Gel or GSM).
A 12volt battery is usually fully charged at 14 volts, but as it discharges, the voltage will drop. Batteries should not be operated at less than 11volts and most likely be damaged permanently if allowed to go below 10.5volts.
If a battery is rated at 100 amp/hours, that means it can supply 100 amps of current at 12volts for 1 hour, or 50 amps for 2 hours or 10 amps for 10 hours.
Now for the more important part. Assume your inverter is 80% efficient. 3000 x .8 = 2400. So the maximum load your inverter can safely handle is 2400watts. At 120volts you will be drawing about 20 amps of current at full load.
If you are going to draw full load, your battery must be able to supply the current to that inverter at an equivalent power rating (power out + power loss = power in). Therefore, 3000watts ÷ 12volts = 250 amps. If your load is only 1000watts (80%), load plus loss will be 1250watts, and therefore, the battery will need to supply 1250 ÷ 12volts = 104 amps. Why is this important? Because the wire between the battery and the inverter must be capable of carrying that current. And that becomes even more important, because if the wire carrying that current is undersized, it could start a fire. Also, with Direct Current, there is more of a loss than with Alternating Current. So if your battery is 20 feet from your inverter, it will incur twice the loss if it was only 10 feet away. You may have to use some very large gauge wire to satisfy the requirements.

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If a normal install, the shore power and generator will go through a transfer switch then to the inverter. If when inverting 120 VAC circuits are OK, then the problem must be before the inverter. I would suspect that it is the transfer switch.

Disconnect shore power, make sure generator in NOT running then take a look at the transfer switch these are in a boat or RV but either can lead to vibration problems which can lead to loosening wire connections which can lead to heating and burning. It may be something as simple as a wire coming loose. (Which could be dangerous!)
Whether or not the contacts in the relays are OK, check and/or tighten all electrical connections. It's a good idea to check and tighten all electrical connections regularly.

Some transfer switches can be repaired by replacing just the relays. Other units (like mine this year!) will need to be replaced entirely.

It sounds like possible bad batt cable or connection as it takes a bit more batt power when engine is cold to crank engine. if you have a volt meter, you can check this. set to volts dc and put one lead on batt neg. post and one on engine block, crank engine, in theroy you should not see any voltage but if you do, that is whats lost in neg cable, in my exp. you will see about .5 volts if you see too much more than that, i would be looking at cable. you can do same thing on pos. side but keep in mind if you go from batt pos. to large stater pos. at stater and you have a ext. sol. like ford likes to use that your loss could be in the sol. contact as well. And under cranking mode your batt should drop to no less that 9.6 volts under load and thats tested at batt post.

Assuming truck is neg chassis/ground

Neg of inverter to neg of batt
Pos of inverter to pos of batt

Hope this helps
Thanks for using FixYa

You may be able to do this but it may NOT charge the external battery, you will have to see.... Anyway simply connect the battery up to the inside terminals Red- Red Black - Black.. with suitable wire & fittings make sure you have no shorts etc. ya good to go.

wire one hot lead to one battery and one neg lead to the same battery

then wire second battery to the first one

your inverter lead goes to the positive post of the first battery

Batteries in the UPS typically last 3 to 5 years and it depends upon several factors. including the number of times the unit must go on battery power and enviromental conditions. There are usually several batteries in the UPS and while the battery voltage may show 13 volts, this may only a float voltage and a true indication of the battery voltage needs the batteries to be tested under a load. there maybe one battery that is faulty and causing the whole battery system to fail.
These lower powered and cheaper KVA type UPS are switch over types (when the mains fail, the UPS switches over to the inverter in milli-seconds to supply mains power). It is a possibility the inverter circuit is faulty and when the mains power fails the inverter does not switch over.

The true UPS type are usually the higher KVA units (over 1500 KVA units) that converts mains power continuously via the inverter circuit. The mains power charges the batteries and the inverter drawing power from continuously from the batteries (therefore there is no switch over time lag from the mains to the battery.

check alt by unpluging batt when car is runing.
check batt wires for losse cont