I think this is a very strange circuit. But if you are learning something interesting in the process, then good for you.
The idea sounds like it should work. In your sketch, make sure all the /OE inputs on the 74hc125 are HIGH before you set either to LOW. This is because if two of the /OE pins are LOW at the same time, the two Y outputs could short out from one to the other, damaging the 74hc125.
You should also have a 0.1uF bypass cap for the 74hc125. And you should have a 200~500R resistor between the Y1/Y3 pins and the first led in the strip. Also a large cap (eg. 100uF or 470uF or 1000uF, depending on number of leds) across the power going to the strip.