Is it possible to reinstall digitizer without damaging it?

Yes, it's absolutely possible. I've done more than 300 iPads that came with Dock Connector issue because the battery didn't get charged. The main iPad 2 flaw is that Apple's engineers failed the exact measurement of the dock connector cable to its logic board port. Being a few millimeters long length of the dock connector ribbon cable, it simply plugs out over any accidental fall or bump. Not the case always but to many many iPad, it happens. It's because the stiff dock connector ribbon cable is bent on its assembly due to extra millimeters length and it produces strong upthrust pressure on bumps. And the cable gets disconnected. So the iPad comes as dead not getting it battery charged. A lot of people have misconception about dead battery and customers are charged a lot on battery replacement during which, the dock connector is plugged back obviously. Actually, Apple was supposed to recall iPad 2 due to its manufacturing flaw, but nobody has raised voice until now. Probably, I'm the first tech to notice it and discuss openly in this community. Hope somebody helps this ahead.

Anyway, in your context, whatsoever is the reason of opening your iPad 2, you can simply open it without any use of heat-gun or specific tool. Simply, get the thinnest and the most stiff Guitar Pick possible. You can get few of them from any music store in a dollar or less. Insert it first from the lower right corner next to Home button. Keep peeling off the edge taking enough time and not prying. Prying attempt by raising one side of the pick breaks the glass easily. A lot of people make mistake here. Keep peeling ahead in a parallel approach. Be careful just above the volume controls because there is a cable trapped between digitizer and frame. If it's a Cellular model, try to be careful peeling to the top of the iPad on the left side of the power button as well. And pay attention to the left lower corner where there is digitizer cable trapped between digitizer and the frame.

You may follow the take apart guide and use guitar pick to get the screen out safely. This is kinda interesting. Hard you may feel because it's your first time, but as you do several of them, it's fun. And you gonna say how funny was the idea of adhesive Apple chose to wrap up iPads! In this process, the best thing is: you don't have to use any alternative adhesive. The same one works fantastic. Just be careful not to touch over the adhesive with your oily fingers. You may also keep it free of dust while working on.

Good luck.