Advice needed: want to buy small transparent LCD panel

I don’t know if such a thing exists but I’m looking for a small (~3"x3") transparent LCD display/panel/thingie through which I can shine a light. I need to be able to control the display via a smallish controller (i.e. not a PC or laptop). It needs to have pretty high resolution (see below).

Background: I bought an Ikea Isbrytare lamp and I want to use it to project (working) clock hands on my wall. The only reasonable way I can conceive of doing this is by shining the light through a transparent LCD screen that displays the clock hands. I’ve made projects involving an Arduino so an LCD that can be controlled by an Arduino would be ideal. I’ve googled around but of course all the hits are for PC monitors or TVs.

I’m also open to other ideas that could project the hands without an LCD. I’ve considered bouncing the light off a clock that has a face made from a mirror (with the clockworks behind the mirror; it would have to run backwards) but I don’t know how to focus the light.

You might be able to hack one out of a broken video projector (the LCD kind, which use a small LCD slide in between the light source and the optics - not the DLP variety, which use a reflective imaging component).

It might seem like a defeat of the object, to disassemble a projector to make a projector, however, you might be able to pick one up cheap if you shop for a second hand unit with a blown lamp - the cost of replacing the lamp is often sufficiently punitive that people dispose of the projector and upgrade to a new, better model.

You may need to consider cooling of your device - the lamp you’re proposing to use is designed to function as room illumination - if you place objects close to it, they will get hot and this could be a fire hazard.

Alternatively, you might just be able to find an LCD clock that already has a transparent display - something like this, or you may be able to hack a non-transparent LCD clock, like this

Thanks for the ideas Mangetout. I was trolling around on the internet and found this page, which has an interesting solution:
The relevant picture
If I can find a backwards clockworks then I can dispense with one of the mirrors. The trick will be finding a clockwork extension like that.