How do you know that the offending driver was drunk?
have you had a look at the CCTV footage yourself and does it show anything?
If you want to claim against the other motorist, you must first identify the car reg no and if at all possible, the driver.
Why not claim off your own policy and let your insurers try to follow up recovery. Mind you, if the other driver remains untraced, that will not be possible.
Consequences of claiming against own policy would be that NCB could be affected.