I've "done" a couple of Burns' Night suppers, for English people, at a village hall club in southern England. I've tried to treat the tradition with respect. So, last year -
Piping in the haggis, with Northumbrian pipes, playing "A man's the man" and "Scots wha hae," and after the first Cut piping it back to the kitchen with "The Mill, the Mill O!"
Last year, I read the poem (shock! horror!) not burlesquing the accent at all. Simply reading what was written there engendered enough of the Scots lilt to satisfy the one who was looking for it.

After the meal we sang the corniest/best known/greatest Scottish songs, a mix of Burns(*) and others, for which song sheets were provided -

My love is like a red red rose *
The Road to the Isles
Loch Lomond
Ca’ the ewes *
Ye cannae shove your grannie off the bus
I love a lassie
Green grow the rashes *
Scotland the Brave
Mingulay
Three craws sittin' on a wa'
Westering Home
Ye banks and braes *
We’re no’ awa’ tae bide awa’

Obviously, a properly Scottish gathering would probably have wanted stronger or less stereotypical fare. My previous experience taught me that the English company did not want to stray too far from the high road.