Powering Raspberry Pi and 12V LCD (HannStar HSD089IFW1) with Battery and AC Input

Hello all,
Full disclosure: I am pretty over my head. I'm no electrical engineer, but I thought, who better to ask than a forum populated by them?
I'm a recent Raspberry Pi owner (for those that don't know, it's a $35 ARM-based Linux computer), and I would like to build a miniature arcade cabinet with it. It would be running Debian Linux (recompiled for the Pi, called Raspbian), which works with MAME and some other emulators. For the monitor, I plan to use a bare LCD panel from an 8.9" netbook (such as the HannStar HSD089IFW1). I've found an eBay seller that sells control panels for these that include the inverter board as well as a 12v DC input to power the entire display (backlight and all). The board has HDMI input, which the Pi can output, so everything on that end looks okay.
My main issue is in how to power the system. I would like to have a setup much like in a laptop or any other portable device where it can be powered both from the outlet and from a battery. The Raspberry Pi uses a 5v/1A micro USB port for power, so I would need both 5v and 12v outputs. I found a Lithium Polymer battery pack that accepts a 12v input and has both 5v USB and 12v outputs, which is great! But, when the battery is charging, it doesn't bypass the power to the other devices like a laptop would, but instead simultaneously charges and discharges to power attached devices, which seems to me like an inefficient solution that would be stressful to the battery.
Is there some way that I could create a system that would a) charge the battery and b) directly power the Pi and LCD control board when plugged in, but c) switch to battery power for the Pi and LCD control board when unplugged? I.e., a system that operates much like a laptop or cell phone would?
I made a sillily rough diagram to help me ask some non-engineering friends this same question
Again, I apologize for my inexperience (I'm a cognitive psychology graduate student-- far cry from engineering-- I just happen to be a gadget tinkerer as well), but I'm hoping to use this as a learning experience for a more fundamental understanding of how devices like this work.
Also, if this is completely unfeasible, could you please explain why that is? I asked this on yahoo questions, and was promptly told to go buy a $200 Dell, which bypasses the whole point of this endeavor, which is to create a DIY project and learn something along the way.
Thanks in advance

There is nothnig wrong with having the charger always connected to the battery. If the charger can not produce enough power the battery will supply the rest. When not being used it will re-charge the battery. To insure that the battery is always charging you would need to know how much your system draws and then make sure the charger is able to handle the max current draw.

So it wouldn't be detrimental to the battery if I had a straight line from wall -> battery -> Pi+display, even if the battery would be constantly charging/discharging when in use in that case?

Sorry if that's what you just said, but that would definitely solve pretty much any problem I have.

no - it would not be detrimental to the battery or charger, you just do not want to over charge the battery so you need a charger rated for the battery you are using.

Awesome, thanks. Really appreciate your help, this info saved me a lot of fumbling in the dark.