Re-download purchased film on iPad without iTunes
I've got to the bottom of this now and there is the potential that some of the information may be useful to others so thought I'd submit the entire thing as an answer.

The reason why the "cloud" icon is greyed-out (and the root of the entire problem) is that the movie in question is No Longer Available from iTunes. I find this bizarre in itself, the movie is 800Mb but for some reason they need to remove that from their servers to save space...?!? 

When the movie was first deleted and we tried to re-download it, it definitely still showed as being available in the store. I guess the metadata was still there but the movie itself was deleted.

This movie cost £10 around a year ago, you can buy it on DVD for less then £5. The lesson here is to have your OWN backups of material from iTunes because they take no responsibility for allowing you to re-download it later. I thought we OWNED this movie and Apple would guarantee our ability to download it onto any device at any time, but apparently I misunderstood that I was paying more money for a product substantially inferior to a DVD.

I found this out by phoning Applecare using the number from Jules' link below.

When going through the "Select an option" stuff, if I selected "iPad", they still wanted £25 to talk to me. By selecting "iTunes", I spoke to them for free. I spoke to a very knowledgeable person who was able to explain the problem and offer - if not a solution, then, at least, a form of compensation.

The offered me 2 movie-rentals by way of compensation, pretty useless for us but better than having nothing for our lost £10.