Hi mlmcholm,
This is what I have.....The Burns you are searching are mostly Irish in origin, fact i.e. born there, such as David's wife, Elizabeth, as I found out on the Statutory Register, as you had already discovered. In the 1871 Cencus David was aged 57, Elizabeth aged 47 and David a weaver to trade. It states on the Census Register that they had an Adopted son named William Haholm, aged two years, a name your 'familiar' with....
All the info: I have on James Burns and Margaret Jardine, is they propose to be David's Dad on Scottish Register..as far as Francis goes, all I have is his name linked with a ? Jardine at the birth of David. I have taken a good run at searching; but the Irish side with the Woolwich interaction etc. with info: in some quarters, missing. I've done genealogy for 18 years and at times it's frustrating. Have you tried the names on the Irish Statutory Register. Try having search on the Irish side, it may pay off on the Burns.
I'm sorry to say, and being truly honest, which one is the Daddy!! Francis or James, I'd rather leave it to you. My friend in Dumfries went for James with the evidence I had given him. I'll give it a breather and take it up later if your still, pulling your hair out.........Ron.