Lord Halifax probably wouldn't have succeeded in negotiating a peace that would have been acceptable to the coalition govt. nor the general populace either. Furthermore, it seems Halifax was more interested in gaining time to strengthen the UK and the Empire, rather than an abject surrender. Additionally, the whole situation at the end of May - June is mired by concerns about the future of France, and whether and what the terms for peace would be. Churchill was of the opinion, that the terms would be too harsh (disarmament), and wanted nothing of it (requesting terms would be seen as a weakness), Halifax wanted to investigate, Hitler doesn't seem to have made them clear, he was probably waiting to hear what was on offer, rather than make a reasonable offer at this time. http://www.bbc.co.uk/history/events/churchill_decides_to_fight_on sums it up nicely. Churchill's entire position could perhaps have been undermined by a reasonable offer from Germany made by 27th May.