IF the battery charger is a switch mode type, and not a mains transformer one, it should run ok from the output of the 3 phase bridge rectifier. Then have it charging a battery. Run your 12V to 240V inverter from that battery.


Have a read of....
https://solar.smps.us/sinewave-inverters.html

As said a few times, get someone who knows what it is all about to help you.
Otherwise you could end up burning the hydro alternator out, and/or damaging your other bits.
There are quite a few different designs in use and if you use the wrong ones, damage will occur.
It is really hard to help without seeing what you have in detail.
And, just out of interest, what is the head of water and the pipe size you are using?

I have this cheap inverter 12V 220V working well, from what you can see you think it can work?

 

After few research I can't find what is a H Bridge Pwm invertor, 1:1 transformer I can find without a problem

https://en.wikipedia.org/wiki/H-bridge

 

OK, if you are happy with your 12VDC to 220VAC inverter but the water turbine and 3-phase alternator is some distance away, this is what I would do:
Run a 3-core cable from the alternator to where your inverter is. This cable will carry the 220VAC 3-phase from the alternator.
You don't say if the alternator is wired as a star (3 terminals) or delta (4 terminals). Does not really matter as only a 3-wire connection is required (on a star the centre connection is not required in this case).
At the inverter end you need a rectifier and a switch-mode power supply to produce 12VDC. Actually, I would add a large 12V lead-acid battery for bach-up so in fact the switch-mode power supply needs to produce about 13.8 - 14V to keep the battery on charge.
This switch-mode power supply will take the variable input voltage and frequency from the alternator and produce a regulated DC voltage.
You can buy ready-made power supplies with a 3-phase inputs (so the rectifier will be included).
Then connect the battery and inverter to the power supply output.
You don't say how much power you are expecting. A single 12V battery (~13.5V) is too low a supply voltage for anything over about 200W. Two batteries (~27V) or 4 batteries (~54V) would be better if you are looking for useful 220VAC power. You would need an inverter to suit these voltages.