I do not know exactly what Ernest Bevin would do at the Exchequer, but whatever course of action he chose, he would argue it with gusto, and he would come to a position of mutual professional respect with the expert professional staff, the eggheads and bean counters, at Exchequer, to whatever degree is possible within his overall Labour commitments. I say this by working off of analogy with his relations with Foreign Office staff.
As for Dalton, at the Foreign Office, he is more likely to be influenced by Attlee and pressures of the median Labour MP opinion, and pressures for consensus among of forces pressing upon him, including foreign ones, above all, his US counterparts.
As such, his positions on Palestine policy are likely to shift more pro-Zionist compared with the previous Conservative government, reflecting Labour constituencies and opinions, and against Foreign Office opinion and 'men on the spot. This will line up nicely with the ongoing Anglo-American Committee of Inquiry, Holocaust revelations, and avoiding frictions with the USA.
The British government would allow Jewish immigration in excess of the White Paper of 1939 limits, in effect setting them aside. Over the course of 1946, the British government would adopt basically the Jewish Agency plan for partition over Arab communal objections, with only minor, tempering pro-Arab Palestinian modifications. Here are two illustrations of the proposal:
Jewish Agency Partition Plan - Map - Hebrew (1946) and
Jewish Agency partition plan for Palestine proposed in 1946 Stock Photo - Alamy.
Basically, the difference of this partition from either the UN adopted partition plan of 1947 or the 1949 armistice lines is that instead of the three Arab-Palestinian chunks of the 1947 partition plan, Galilee, West Bank, Gaza and west Negev, and the difference from the left-over West Bank and Gaza Strip from the armistice, is the Jewish Agency plan left no Galilee or Gaza Strip for the Arab Palestinian side, but just a mega-West Bank, and a port outlet at Jaffa, with or without a corridor. And Jerusalem would be an international enclave.
By allowing immigration without restriction, and accepting the Jewish Agency side's partition concept, the British government essentially would be at peace with Haganah, the largest, most powerful Jewish militia, which would not need to spend energy ingeniously circumventing British immigrant interception operations. But the Revisionist Irgun and Stern Gang militias would reject the solution which fall way short of their desired borders of including the whole Palestine Mandate and Transjordan and keep up assassinations and bombings. However, their effectiveness would be kept limited in scale and sporadic through British administration and Haganah cooperation. However, Arab Palestinians would demonstrate and conduct violent attacks in protest of partition. Here, in addition to some limited # of vetted Arab Palestinian policemen the British administrators could rely on, while others quit or moonlight or get fired as security risks, the British are relying on the Emir of Transjordan and his Arab Legion to ultimately occupy and administer the allocated Arab section of Palestine.
This all increases the anger and bad press Britain gets among the Arab Palestinian press and in Arab-majority countries, and reduces the bad press and negativity Britain gets in Jewish sympathetic outlets throughout the western world in Europe and North America. The French, out of a vindictive, 'get-even' policy against the British, and still angry about getting ushered out of Syria in summer of '45, may still support, as historical, Irgun and Stern Gang terrorists, but possibly also Arab Palestinian terrorists who target the British as well.
Overall however, in this ATL, the British in the late 40s are facing a far less formidable security challenge in Palestine, suffer fewer losses and humiliations [they don't get the King David Hotel or Acre Prison blown up], and exit their forces with greater smoothness from the mandate by agreements with Israel and Jordan, without deferring the whole Palestinian question to the UN for ultimate advice or solution.
The lack of 'seams' in the British withdrawal, means Britain hands all border posts of the Mandate with Egypt, Lebanon, and Syria to Israeli forces in an organized manner, deterring the former three countries from mounting an invasion. And silent truce (at least silent on the Jordanian side), prevents the outbreak of fighting between the Arab Legion and Israeli/Haganah patrolled Arab/Jewish zonal boundary.
A consequences of this is that Israel and Jordan both declare independence as members of the Commonwealth of Nations, and their independences are not accompanied by forced displacements, except for the Jewish evacuation of some settlement blocs. Jerusalem remains international with a British garrison.