The responses so far have squeezed as much as can be obtained from on line records (and yes we also wish our archives were free to all like others!).
Without his record, which would include both his other rank and officer service, we don't know which element(s) of KEH he served with (although the deaths of three 2 KEH men in Ireland that Jervis42 cites does indeed suggest that was his unit at that time at least). The regiment has a complicated war history as you will have seen for the site that iluleah cited.
This National Archives link shows the war diaries and also illustrates that the regiments operated by squadron attached to different divisions at different times:
http://discovery.nationalarchives.gov.uk/results/r?_aq=king%20edwards%20horse&_cr=WO%2095&_dss=range&_ro=any&_p=1900&_st=adv(Part of one of the war diaries for 2 KEH in 1915 (before he was overseas) is in the Canadian Archives (access free) because they were attached at the time to the Canadian Corps).
Thus we don't know which war diary to look at for the chance of his name appearing although, as his medal card shows that he didn’t serve overseas until Jun 1916 and 2 KEH was disbanded in August 1917, one or two could be discounted.
All that said, your interest in your man in the 1916 Ireland period is likely to be difficult, dare I say impossible, to pin down. His record will have nothing about it. Not from a secrecy point of view but as Ireland was part of UK at the time, his time in service up to when he went to France in Jun 16 will simply be covered by a heading “UK date from/ to”
You would be best advised to seek out more general sources rather than an individual’s service record. Philip has pointed you to “The history of King Edward’s Horse (The King’s Overseas Dominions Regiment)” edited by Lieut-Col. Lionel James (London: Sifton, Praed & Co., 1921). I have seen discussion about the composition of the firing squad that executed the leaders of the uprising over 9 days in May 1916. The book “The Easter Uprising” by Foy and Barton records that the squads were found by the 59th Division stationed in Ireland at the time and the weight of research indicates that they were from the Sherwood Foresters. As Jervis42 has pointed out, he was not commissioned until almost 2 years later so he would not have commanded a firing squad and, while he was in Ireland at the time, I would be sceptical about him taking part without contemporary evidence.
MaxD