If I was accompanying the tune the pipe on the hob, i prefer standard tuning, because it is easy to emphasise clearly that the second part is in d major using d major chords, for the first part I would use long d, a d modal chord that is easy in standard and then go to c major, it encapsulates perfectly the switch between modal and major.
or alternatively do not play top e string for the first d modal chord.it is not necessary to play all six strings all the time[something basic guitar tutors rarely point out]