Quoted from Rensh:You can make this much more easy
Just take
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Put in front a bridge rectifier and connect the input of the rectifier to your GI circuit. The rectifier will make it more or less DC and the step up converter will make it 12VDC required to drive the led strips.
Sure, if the GI circuit can handle the extra load of also driving the LED strips. Maybe if all the GI bulbs were replaced with LEDs there might be enough spare capacity. I'd measure the power consumption of the LED strips first before trying that approach.