Was there any particular shame attached to losing the war?
They genuinely were not respected by the greater society, the war was very unpopular at home since many British subjects resented having to pay higher taxes to fight a war an ocean away to kill British subjects. Britons in Britain hated the high taxes, and the government which many felt did not even represent their interests let alone the colonies and there were numerous riots that had to be suppressed. Most of the common soldiers were either recruited from the lower levels of society, or outright impressed for petty crimes such a vagrancy. Many went to serve on in other parts of the Empire but they did not come home to parades or cheers from thankful citizens. As for the officers that fought in the Americans it was generally seen by them as blemish on their careers, a war which even officers like General Clinton thought was wasteful and counterproductive was not something they wanted to be remembered for. The defeated career officers were in the end lucky as the war was quickly forgotten in the public mind because of the great two decade war against Revolutionary France that followed which brought eventually victory and a tenacious nationalist spirit that helped unify British society. A good number of the heroes of the Napoleonic wars began their careers fighting against the Americans, but that was rarely an element of their service that they were proud of.
The Men Who Lost America: British Leadership, the American Revolution, and the Fate of the Empire Andrew Jackson O'Shaughnessy