Hi dB C,
Ordering a new LCD might not help.
Since you are replacing the battery and the LCD was good before, replacing the LCD would not help. The problem is with the Zif connector where the LCD ribbon cable plugs in. If it does not have good contact, it would not work.
From my own experience, I tried to connect and disconnect the cable a few times using different mylar pieces. I was ready to give up when by pure luck, I applied some pressure to the area where the connection located and voila, the screen turned on.
I would suggest to try it again with different "tape/mylar".