In your position I dont think it matters- go for the one which suits. The three providers you have listed are much of a muchness, and will be regarded as so by the vast majority of recruiters. Magic Circle firms tend to send their prospective trainees to the same college because it makes it easier for them to (a) keep an eye on them (have heard that A& O trainees on the LPC get a slap from their paymasters just for missing classes!) and (b) make the provider adapt the course to suit their individual needs. To be blunt, you probably dont have much chance of securing a TC with a firm with such requirements. The majority of leading firms outside the MC (Think Cleary Gottlieb, Macfarlanes, Jones Day etc.) dont appear to give a toss where you train.
It is also useful to note that most of the big city firms (if thats where you want to end up) will be slightly reluctant to consider an applicant who hasnt pursued the traditional route. Local firms will probably take where you did the LPC as an example of how committed you are to their area- if, say, you intend to work in York but do the LPC in Leeds, York firms might also be a little suspicious if you claim to have no desire to work anywhere else.
Hope that helps.