Depends where you want to work, don't know much about London airports, but there is Aviance as well, and then whatever local companies there might be at LHR or LGW. I have to say that I am not even associated with despatch, although my brother in law has been a despatcher for 10 years or so, and has worked from nearly all of them at MAN.
I do have to deal with the handling compaines with my job though, and having been at the airport for 8 years now, reputation goes a long way, that is why I can look at Circusair with a smile on my face. Incidentally, do you know they have either bought out, or been bought out twice in recent years? Hmm, stable firm that one. The pay is lousy, T&C are below par and although I appreciate you only wanting to do 20 or so hours there is a problem. You strike me as someone with a passion to do the job, and give it your all. If this is the case, go to a company that will like you for it, and will reward you for going the extra mile.
My impression of Curcusair, having recently dealt with some ramp managers, is that they promote the pains to get them out of the way. I have never in my life witnessed as much "buck passing" by one person than I did the other week. Beware, if you are good at the job you will stay doing it!
For my money, go to Swissport, a very proffesional outfit from what I can gather (others might disagree) and are likely to reward a good worker!