I was turning the display off because I did not want the light from it to deter the potential subjects. I hope that is what you mean by "live view". In any event, the camera was turning the display off after a while anyway, and I have to override that with CHDK to get it to stay on. If I have to leave the display on I will have to stick some sort of cover over it, but I can do that.
I have done a couple of quick tests based on your suggestions.
Camera had been waiting for the trigger for around 11 hours. Before triggering it, I turned the display back on and the response time was 3900ms. Having done that, I don't know if the response . would have been slower if I had left the display off, but 3900ms is in the sort of range I have been seeing - I have seen 4718ms when the camera had been waiting for 24 hours.
I then triggered the camera a few times. This was without leaving it waiting for an extended period of time (and for which I was getting responses of around 2200ms before with the display off). I got a response of around 1350ms, so it looks like leaving the display on does help. It probably also explains some of the variability in the results I got previously, where I did see responses of around 1350ms, it is quite possible that I had left the display on while I was testing, without realising that could make a difference.
So far, so good. I then set the white balance to custom, and tried triggering it again without leaving it waiting for a long time. (Camera is set on "P", manual focus set). Three times I tried to trigger it, and three times it failed to take the shot. It was reacting in the same way as when I had set the camera to "M" and set the aperture, shutter speed, and flash power: the camera does react, the auto focus assist light comes on briefly twice, and there is a tiny mechanical sound, but the flash does not go off, and no pictures were taken. I get the impression it is trying to tell me something, but I don't know what it means.
I set the white balance back to auto, and could trigger it again. So I tried setting it back to custom and tried again. First time, same double blink of the auto focus assist light, no picture. Second time, and wow it took a shot - response 1304ms. I have no idea why it started working now. Triggered it two more times, responses 1345ms and 1346ms, so, if does not seem changing the white balance to custom makes much of a difference.
I still seem to get the effect that the longer the camera has been waiting, the longer the rseponse time becomes - I got responses of 1839ms and 1739ms when it had been waiting for several minutes. I will leave it for a longer time and see what I get.
In summary, what I have seen so far is that leaving the display on does help, but setting the white balance to custom did not make much of a difference. Setting the white balance to custom may also have induced a re-occurrence of the problem where the camera fails to take a shot when triggered, but the local jury is still out on that.