Hi John
If you get no joy here, you should try a post in a general section. All you need is someone with an Ancestry account, as you can drill down - 1871 Census, Lancashire, (Manchester), Hulme - and you should see the barracks as a named district. If the numerator did not record the name of the regiment, you should post the name of the officers back in this section, and someone can look them up in an Army List. Officers are usually listed at the front of the census schedule.
Also, if you can get to Kew, you can check to see if the man's papers survive. Papers from this period are filed under Cavalry then alphabetical, so you do not need to know his regiment.
Ken