Can I connect a cooker hood from the battery?
He will give 2-2.4 A of current from the force, with an output voltage of 5V - this is 10-12 watts of power. The power of conventional hoods is 150-200 watts on average.
Do you think that 12 watts, at least for some reason, will be able to pump the coveted 400-600 m³ of air? I believe that under no circumstances. It’s not just “weak”, it’s nothing at all. It's like you plug in a couple of coolers from a PC ... Honestly, don't even try. If we are not talking about a bank, namely, a battery from a telephone, even more so. They have a small current return.
If on business, then here you need to come up with not ready-made solutions, but assemble them from separate modules. For example, an assembly of lithium cans of the 18650 3s format (3 in series, parallel until the desired capacity is achieved), by the way, they are used in the bank, but here you need to use not 3.7V - 5V, but a normal inverter 12-220 to 400-500 W and feed from this assembly. But they will sit down relatively quickly, constantly charge, it either remove them or pull the cable with the charger to the installation site. You can use a car battery or maintenance-free (as on uninterruptible power supply units), but the essence will not change.
You should not spend money comparable to the challenge of a competent installer-builder.
What is the problem, even with repairs made, breaking into the chimney to connect a NORMAL hood? Any decent wizard will do this with minimal damage. And if they are inevitable - buy a built-in cooker hood and put it in a beautiful hanging cabinet. Lay the cable for connection behind the baseboard and neatly behind the furniture or somewhere in any other convenient place to position the socket for its connection. You can lay it yourself, there are even special skirting boards with a channel for laying the cable. If you don’t know how to do it yourself, then a NORMAL (not a fraud) electrician will take about 1000 (+ - 500) for this.
So you get a GOOD solution that you will use and not remember either the battery charge or regular purchase of new ones.