With an APU as a deferred defect, shouldn't you dispatch with a proviso that your destination(s) have been advised to provide you with ground power and air? Maybe not - but in my experience, we would not shut down the last engine until the previous engine had reduced below a certain N1 rpm. Ground air cart would then have taken over as 'motoring air'.
On a previous fleet- L1011 TriStar, with the early RR RB211's, tailpipe fires were not infrequent, and with spectacular visual displays!