I think you're on the right path with DLP idea. There are DLP sets that use HID or High pressure discharge lamps that are broad spectrum, but also have UV content. There will be UV cut filters in the optical path to prevent premature aging of components, organic die based filters and the polymer based back illuminated screens (if a rear projection unit). Also the UV filter is in there to protect human eyes. It will be a simple matter of removing the UV cut filter, although I recommend you wear UV protection goggles until you can measure the spectral content. It might be possible to just replace that single element to provide wavelengths you need (rather than removing it wholesale).
Do note that the index of refraction might be quite different in the optical path so the UV image will be out of focus relative to the visible range. I am assuming of course that these mice are NOT highly trained so they won't be able to provide relative IQ (Image Quality) metrics. So perhaps you should wait until this training happens. Or perhaps I should say possibly as these particulars will be dependent upon each units optics design, the glass used and lamp used.
Also be aware of flicker issues, with the mice. I presume that their response time is much faster than ours so they may find a DLP as being too flickery. The DLP flicker is noticeable peripherally in some humans so I assume that mice may avoid this - i.e. movement based triggers and alert in the visual cortex. evoking Run away! or Freeze! might mess up your experiments.
As an aside we used to take the UV filter out of our test setups (LC based systems) to accumulate total UV dose on the LC system to be able to study long term degradation and to predict MTBF and lifetime and we used similar lamps to these DLP units.
Make sure the little guys have good sun screen.