A flight of fancy, if ever there was one. Probably as valid as claiming that it is a parable of how church and state depend on each other.

I can see three probable explanations for the laughter.

1) BA is the most callous and cruel creature in balladry.
2) BA is demented.
3) BA suffered a nervous breakdown.

#3 makes the most sense to me, personally.

As for the bloody shirts, an allusion to the Gospel of Luke is a more reasonable assumption. 22:44 "And being in an agony he prayed more earnestly: and his sweat was as it were great droops of blood falling down to the ground."
If one prefers a medical to poetical explanation, then what about bleeding ulcers or vomiting blood? Admittedly, I'm not a doctor, so could be wrong.

I've found Robert Graves' comment (it is in "English & Scottish Ballads", 1957: "It is clear enough that Sir John Graeme did not die merely of a broken heart. Like Clerk Colvill, he seems to have been a landowner who had an affair with a country girl, but later decided to marry a woman of his own class. When this marriage was announced, the girl avenged herself by bewitching him; the procedure being to model a wax image of the victim, make it more real by adding his own (stolen) hair-trimmings and nail-pairings, and then gradually waste it over a candle, sticking pins into parts that the witch wanted to injure most". I don't know whether Robert Graves is right or not, but somehow we must try to explain Barbara Allen's laughter when she sees his corpse and all the verses about the bloody shirts, the watch, the basin full of tears etc. I've chosen some of these verses from some of the most beautiful versions of the ballad I know:
From Martin Carthy's:

And look at my bed-foot", he cries,
And there you'll find them lyin',
My sheets and bloody shirts,
I sweat them for you, my Ellen."

She walked over yon garden field
She heard the dead-bell knelling
And every stroke that the dead-bell gave
It cried, "Woe be to you now, Ellen."

As she walked over the garden field
She saw his corpse a-comin',
"Lay down, lay down your weary load
Until I get to look upon him."

She lifted the lid from off the corpse,
She bursted out with laughin',
And all of his friends that stood round about
They cried, "Woe be to you now, Ellen."