The shift register idea is quite clever. I do see a potential problem however. The use of a 1N914 diode to hold the LCD module E line low whenever the shift register 6Q output is low is compromising the low level noise margin of the E signal. If you look at the web specification for a typical character mode LCD module such as here you can see that the spec for V_IL is 0.6V max. It is rather hard to meet this specification with a diode in the path that drops 0.6 volts plus whatever the output V_OL of the shift register is.

If the diode must be used I would suggest selecting a low drop Schottky type and then just for a hobby application of the circuit. If the circuit is intended to be used in a production situation or where the cable to the LCD module has any length from the shift register then I would use the "AND" gate as suggested by the referencing WEB page. Here is a convenient type of gate that could do the trick nicely...single gate AND.

Michael Karas