I just got a call back from the RSPCA. The woman I spoke with is bringing round a humane trap either this evening or tomorrow, hopefully, and she'll be showing me exactly how it works and everything so if it happens to catch the wrong cat, like one of ours, I can release it.
I did ask about their policy on putting cats to sleep, and she told me that they only consider that if the cat has FIV or something equally serious that could be fatal to the cat, or if they have something dangerous that could be passed on to other cats. It did seem like putting the cat to sleep is very much a last resort option, as she said they do try to re-home the cat if at all possible, which I think it is with this cat.