Your concept is good although you will most likely need higher current output from your transformer.
And you can omit the capacitors, voltage regulator and single diodes. Batteries are not fussy about the charging waveform. Incidentally a 12V regulator would only charge the battery to 12V and no higher. It ought to go up to 14 or 15V by the time it's fully charged.
The led can stay since it's an indicator.
Your battery is 150 AH. To charge it in, say 15 hours, you'll need 10 amps going into it for all that time. This is a C/15 rate which is typical.
You can use an 18VAC transformer and it will still work.
Even an 11.5 VAC will deliver on the peaks. And it will automatically taper the charge as the battery reaches full capacity. Charging will automatically stop at VAC * 1.414. Minus 1.2 V diode drops through the bridge rectifier. This equals 15 V. A typical end-of-charge voltage for a lead-acid battery.
After all that... it may be more practical just to use an automobile battery charger.