Searched two different Belfast districts manually (page by page) but still no sign of Annie/Anna. Never found David's birth (c1903) either.
No match for William McClurg in CWGC database (I search just 'McClurg' and there aren't that many). No mention of an alias in pension record?
Which brings me back to Annie's birth. I think Army would have wanted proof that the wife was actually his wife and children were his children before paying out a pension. I have seen similar cases where children were listed then crossed out as they were step-children, etc. Minnie could, if needed, have submitted her marriage lines (we don't know outcome of bigamy allegation) but Annie was born in 1909 which is before the marriage in 1910.