You've correctly identified part of the problem the British faced - public opinion in other countries, particularly America, seems to have been against the British conduct in Ireland. In fact, we have Lloyd George's (the British Prime minister at the time) memo to his cabinet about the signing of the articles of the treaty, which justifies their thinking behind giving Ireland "Dominion" status (putting it on par with Canada, New Zealand, and Australia) and they specifically mention relations with America (the emphasis below is mine).

"... a permanent settlement of the Irish problem would enormously increase Great Britain's prestige in the world and would show that she was still capable of over-coming almost insuperable difficulties A just and righteous settlement of the Irish question was vitally necessary at the present moment and would have the most beneficial effects in America and elsewhere."

So as we can see, the British government certainly had their reputation in mind when they agreed to ceasefire and talk with the Irish provisional government. American public opinion took a rather dim view of the behaviour of British troops in Ireland: the reprisal burning of parts of Cork city by auxiliary troops in 1920, for example, led to the foundation of the American Committee for Relief in Ireland (ACRI), in their 1922 report, they claim that "Committees were formed in every state in the Union to carry on the great work of humanity." (pg.48). The organisation sent an investigatory delegation to Ireland, on on foot of which the ACRI ask for, and got, donations totaling over $10 million.

Apart from America, the Irish situation was being reported on with interest elsewhere in Europe. Quite smartly, the provisional government (or "first Dáil") had a network of envoys and publicity agents in other countries in Europe, anticipating the need for for a propanganda war side-by-side with the physical war. Many foreign journals sent special reporters to Ireland, their visits facilitated by the people like Art O'Briain and Desmond FitzGerald, the first Dáil's London representative and head of the Publicity Bureau respectively. French press was broadly sympathetic to Ireland, the front page of Le Petit Journal, a popular daily paper, carried a full-page depiction of Terence MacSwiney, the Lord mayor of Cork, attended by a priest in his hospital bed on hunger strike, captioned "Le martyr Irlandais" (The Irish Martyr). Similarly Il Popolo d'Italia (edited by none other than Benito Mussolini) carried the headline "MacSwiney agonizza.... Viva la republicca Irlandese!"

MacSwiney was a particularly popular figure abroad, he had been elected to the first Dáil in 1918 and as Lord Mayor of Cork city in 1920 (his predecessor was assassinated by members of the Royal Irish Constabulary, the civilian Police force). Macswiney was sentenced to 2 years in Brixton prison after being found in possession of seditious material. His hunger strike drew great interest and his death 74 days into the strike inspired political cartoons in British, French, Spanish, and American papers, many of which implicated the British authorities in his death.

Even British press was not entirely behind the actions of their government and military. Lloyd George was apparently informed by the owner of News of the World that the public were "sick and tired" of hearing about what was happening in Ireland. Several newspapers, such as The Manchester Guardian and Daily News made a point of reporting on the British strategy of punishing civilians for the actions of the IRA. I've already mentioned what was done in Cork, but the town of Mallow was also partly destroyed in a reprisal attack, and in a separate incident the military killed three railwaymen in the town after an IRA assassination attempt on an RIC captain. In Balbriggan, North of Dublin city, 50 homes and several business were looted and burned and two men were beaten to death after two RIC members had been assassinated in the town earlier in the day.

Incidents like this started to put pressure on the British government - the Balbriggan incident led the leader of the opposition H. H. Asquith to compare it to the actions of the German army during the rape of Belgium in World War 1. The large Irish population in Britain and people sympathetic to the situation organised massive protests, the Irish Self-Determination League (ISDL) boasted membership of 27,000 members in late 1920, rising to nearly 39,000 in March 1921.

In tandem with public opinion, the British forces in Ireland were not having an easy time, I won't go into the military side of the struggle for independence, but I think this quote from General Macready in mid 1921 sums it up:

"...Does the cabinet realise what is involved? Will they go through with it? Will they howl when they hear of our shooting a hundred men in one week?"

Similarly, Lloyd George's man in Dublin Castle, Andy Cope said:

"We are willing to acknowledge that we are defeated... There is nothing else to do but to draft... four hundred thousand men and exterminate the whole population of the country, and we are not willing to do that."

So as you can see, the First Dáil and the IRA waged not one, but two successful campaigns, one with bullets, and one with public relations. The British government was forced by a combination of internation pressure, internal pressure, and military pressure, to do something to bring the fighting to an end.