All this assumes Hitler offers reasonable terms, which is very unlikely. His MO at the time was to dictate harsh open-ended requirements. Earlier his track record was of violating agreements he had declared permanent and irrevocable. Whats going to happen here? Halifax agrees to something that at best looks like the Munich agreement and he repeats the line about peace in our time?
That's certainly true, but shoulden't we also consider the circumstances in which he'd been operating up until now (Specifically, the state of the states he was 'negotiating' with) vs. his situation relative to Britain? In his prior invasions, a combination of the German army and cracks in the popular and government morale resulted in him dealing with nations that had been rendered and believed themselves to be prostrate, which Hitler and his generals knew lacked the ability to meaningfully "fight on". What resistance they would be able to offer would be disorganized and minimal, and would result in far more damage to the resisters and their family/neighbors themselves than they could ever hope to inflict on a German occupation force. As such, it only made sense to dictate hard terms.
Britain, on the other hand, still has a military with morale and organization. The Royal Navy and Royal Air Force are more or less intact (Even if, as the BoB goes on, the later gets tired and somewhat overstreched) and the Home Guard and elements of the British Army prepared to defend its shores. Any attempt at Sea Lion, even if successful, was universally agreed to be more damaging to German military power than British, and if a failure would be even more problematic. In this situation, therefore, there's actual benefit to presenting more lenient but more acceptable terms to London than to Paris, Copenhagen, or Warsaw: in the later cases, being lenient is a net lose for Germany by leaving in hostile hands resources/power that could have been taken/subdued at little to no cost. For Britain, getting peace via diplomacy has the potential to be a net and comparative positive to Germany vs. rendering Britain prostrate, which would require expending a great deal more resources to even have a chance at being successful.
I'd imagine Halifax, were he Prime Minister, would make something like the following his internal policy: If Hitler approaches us with actually honorably terms, we'll consider accepting a cease-fire at least (If only to get time to co-ordinate with the resistance, consolidate Imperial resources, woo The United States, ect.), especially if it would give vital breathing space to mitigate the threat of a "stab in the side" from the shadow looming over Asia. If you can get him to publicly lay out demands that are unreasonable, on the other hand, than you have concrete evidence with which to shot down the radical doves/defeatists, can be more certain of rallying universal public opinion and thus prevent the risk of "Better Hitler than Blum"-type sentiments from undermining the war effort, and get a stronger diplomatic card to play with Roosevelt, Stalin, and the heads of the Domains.