This isn't a particular specialty of mine and I haven't looked at this particular question before, but the following passage from William Roger Louis' The British Empire in the Middle East 1945-1951 seems to provide precisely the answer you're looking for, and I can also provide some background context afterwards. Bit of a long excerpt but I think it's a good explanation:

[After the armistice] No one wished to be associated with anything that implied a permanent settlement. But the upholding of the armistice lines seemed to be better than the risk of lapsing back into war. Thus the three powers [Britain, France, and the United States] declared that they would oppose any attempt to alter the armistice lines by force. The actual territorial issue being debated by the British and the Americans was the question of the West Bank. The longer the status of Arab Palestine remained indefinite, the greater would be the danger of an extremist independent Arab state led by the Mufti that would be hostile to Jews, British, and Americans alike. It would become, in the British view at least, a hotbed of Arab fanatical nationalism and irredentism. Both Britain and the United States therefore backed King Abdullah's annexation of the West Bank on the 24th of April 1950. Almost a month to the day later, on the 25th of May, the three powers issued the tripartite declaration. In that sense it served as a guarantee of Jordanian annexation.

'The US and the UK were basically after the same thing in this matter', observed on of the Middle East hands of the State Department, Raymond Hare, 'and it was purely a question of tactics.' Both the British and the Americans wished to prevent the deterioration of Arab Palestine, and, again in Hare's words, 'to let an explosive situation quiet down.' Both the State Department and the Foreign office viewed Jordan's annexation of the West Bank as a step towards stabilizing the region, and the Americans as well as the British saw one of the purposes of the Tripartite declaration as a restraining influence on the Arab leaders who objected to the expansion of Abdullah's kingdom. When all is said the Americans were more inclined to accept the 1949 frontiers as a permanent demarcation than were the British. Nevertheless the talk about Jordan's 'corridor' through the Negev to the Mediterranean had long ago evaporated, not least because of American insistence that the Israelis simply could not be persuaded to relinquish any substantial piece of territory won in the 1948 war. The State Department now found itself tacitly if not openly supporting the territorial frontiers of Israel as they had been carved on the fields of battle, but it is equally important to emphasize that the American documents indicate a sensitivity to such issues as the sale of refugee property within territory under Israeli control. As for the upholding of the armistice lines rather than pushing on towards a general peace settlement, a statement made by Raymond Hare summed up the American view, which was remarkably similar to the British estimate: 'It is our belief that, motivated by his own personal interests, King Abdullah has gone as far as he could in attempting to reach an agreement with Israel.'

...

Furthermore [Bernard Burrows of the Foreign Office] believed that the Zionists basically misconceived the Arab reaction to the territorial claims [of Israel], and for that matter the British ability to reconcile the Arabs to the new Jewish state:

Israel's major need is peace with her Arab neighbours but she greatly underestimates the difficulties in the way of securing it. It is a myth that the Arab peoples are not opposed to Zionism or that the idea of Arab unity is an artificial creation: Israel estimates of our influence over Arab leaders are also largely mythical.

It was the last point about British control of the Arabs that was critical on the eve of the tripartite declaration. 'The influence of the British government upon King Abdullah', Burrows told his American colleagues, 'was not unlimited and...the King seemed definitely determined to go ahead with a formal proclamation of union' with the West Bank. To prevent the 'disintegrating tendencies of Arab Palestine', Burrows emphasized, ' the psychological stimulus of formal incorporation [of the West Bank into Jordan] was necessary'. In that sense the tripartite declaration, by endorsing the armistice lines, may be viewed as the best solution that could be found at the time to the problem of the West Bank.

I think that gets to the core of why Britain acted is it did, which is that in part your question takes for granted a few things that weren't necessarily true at the time. For one thing, a "controversial" stand that effectively united Britain, the United States, and France (even if the latter two didn't entirely explicitly "recognize" the annexation) could hardly be said to have isolated Britain in global politics.

For another thing, Jordan was a British client. As is referenced in the bit about the Jordanian "corridor" there had been intimations by Britain to King Abdullah from the time of the establishment of British Mandate for Transjordan in 1921/22 that at some indefinite date in the future that some or all of the directly controlled territory of Palestine might be ceded to him.

One of the reasons for the Arab failure in 1948 is that each of the main sides territorial ambitions for Palestine, all of which had some degree of historical plausibility (Palestine having been historically part of "Greater Syria", "Transjordan" merely being the eastern half of what had historically been a unity of the east and west banks of the Jordan valley, and Egypt having long been the Suzerain of the Levant and having actually conquered it in the 19th century).

The result was that while only three countries explicitly recognized the annexation (and actually this is disputed, see this note on Pakistan's supposed recognition. Some sources appear to say "only Britain and Pakistan" others "only Britain, Pakistan, and Iraq." Britain's recognition appears to be the only entirely non-dubious example) only Egypt and Saudi Arabia called for Jordan's expulsion from the Arab league in response. The rest of the Arab league effectively declared it to be a final status issue, neither recognizing nor really condemning the annexation. As a result they accepted annexation as "a practical consideration" for the Jordanians to "hold this part as a trust until a final settlement of the Palestine question [was] realized...."

For further background reading and the source of the quote immediately above see D.K. Fieldhouse Western Imperialism in the Middle East: 1914-1958.