There are a number of different reasons, a few you highlight above:

Reputation/Perception - Sending in soldiers would have been an open deceleration that the United Kingdom was in the middle of a civil war. It was very important for the British government to frame the Irish nationalists as a small, disloyal, poorly supported criminal element which would be handled by the police. They were able to get around this by creating "policing" units like the RIC Reserve which were demobbed British soldiers. Soon to be known as the "Black and Tans" due to their mix of police and khaki military uniforms they carried out a campaign of government supported reprisals, such as the burning of Cork City. This caused considerable outrage in the British press. The President of Sinn Féin and Dáil Éireann, Eamon de Valera, was in the United States at the time, and was leading a propaganda and fund raising campaign. Resolutions calling for Irish representation at the Paris Peace Conference had been passed by the House of Representatives and many of the heavy handed acts of the British in Ireland were highlighted. Sending in more soldiers would have caused further reputational damage to Britain.

Changing face of the Empire - Ireland wasn't an isolated case. There was unrest in Egypt, India and Palestine at the same time. The expense of suppressing resistance to British rule in these areas was too much, especially considering the cost of the World War. Empires like France and Britain had now entered a period of decolonisation and began to realise that they could no longer hope to rule these areas with force. Britain may have been able to deploy overwhelming force to crush the Irish Nationalists but at great financial cost. After that would come the cost of rebuilding the country and maintaining a military presence. It would also just kick the can down the road; how long before the next outbreak of violence? Britain realised it would need to compromise and give the Irish a measure of self government. Make them invested in the success of the Empire, slowly morphing into the British Commonwealth, so they themselves would maintain peace.

Neither side could win - During The Troubles, in the mid 1990's, the British military reported that the Provisional Irish Republican Army could not win the conflict. However, neither could they. Various splinters of the IRA had kept the conflict going for 30 years and could do so for another 30 years. With only a tiny core of fighters and support from a fraction of the community they could carry out attacks with massive human and financial cost. The same realisation was made in 1921. As an irregular paramilitary organisation, units of the IRA could have survived and carried out attacks even if Dublin was overran with British soldiers and the entire Dáil Éireann government arrested. Short of destroying villages and towns and resorting to Boar War style concentration camps they would never be able to fully defeat the IRA. Neither side could win but continuing would cause financial and reputational loss to Britain and the cost of victory was too great.

Peace feelers were put out by the British in late 1920. The Irish responded so positively to these that the British Government believed that they were on the verge of collapse and redoubled efforts to defeat the IRA. The Irish were severely stretched at the time and probably would not have last much longer but were able to pull off a number of spectacular attacks such as at Crossbarry in March 1921 where 100 volunteers engaged over 1,300 British soldiers. In May 120 members of the Dublin Brigade attacked and burned the Customs House. Most of them were captured but the scale of such attacks led the British to believe the IRA was nowhere near defeated and a truce was agreed starting the 11th of July.