Hi
Richards advice is, as usual, great. You wont get a London sound eng for that budget especially carrying 4 radio channels.
Id add that, if you feel that your clients expectations and requirements are above what you can deliver, either ability wise or budget wise, you should always be willing to WALK AWAY. If you go ahead, fail to deliver then they will more than likely be un sympathetic.
Def lav mic on every person. You can hire wireless mics in London very easily and they can be plugged directly to your 4 camera inputs available but
1) youll not be able to monitor all channels so bear a risk
2) have a large job in post edit to sort out the audio and close the non used mic channels or live with horrible colouration.
3) accept that the 2 camera audio file will NOT sync up exactly in your edit timeline as the exact moment will vary when you press record on both cameras, so the exact frame start will vary up to 1 frame causing a metallic feel with more mics left open than the one being spoken into.
Id consider WIRED lav mics if people can cope with movement being restricted. ECM77b s in London are about £15 per day!
Peter
Post Production Dubbing Mixer