Is there really no room for lights? A stand-mounted powerful speedlight with a wireless trigger will add quite a bit of light, with a reasonably small footprint. A 360 diffuser will have the effect of increasing the light by bouncing it off all surfaces. I have done similar in the past, and was recently at an indoor event where the photographer used this solution.
The speedlight (or larger unit like a Sunpak 622 Pro or Metz 60) can be positioned near camera position with a diffuser or bare bulb. Another option is to position the light to camera left or right, with the normal, directional, head pointed at the subject. The first solution is a more ambient and softer effect, the other is more directional and harder.