My understanding of the rule is that the APD type for a departure from the UK is based on the location of the 1st stopover point and the class of travel

Given that the 1st stopover point is in the USA the standard rate of GBP80 is due. If the whole journey was in the lowest cabin for the whole journey to the 1st stopover point, then a reduced rate of GBP40 will be charged. Originally the journey was in economy class and so GBO40 was charged; to upgrade part of the journey to a higher class of travel then the higher rate will apply. If a stopover is put in at BOS, then the issue should go away. If there is no change fee, then perhaps change the bos-mia to be a day or so later and then try changing the date back again and see whether the tax requirement is picked up

Dave