I always thought that simple tax paid by the airline or operators based on a per-seat charge would be easier to administrate, and allow the airline to pass the charge on in full, in part, or not at all, or allocate different proportions of it to different areas of the cabin, in effect charging this at the front who have paid more for their seat more in money terms but less in percentage terms based upon the amount the passenger had actually paid for the seat. e.g. A passenger paying £300 for their seat, of which £ 30 was allocated as APD (or whatever it would be called) would pay substantially less in percentage terms than another passenger, travelling in cattle-class for whom £30 is a huge percentage of their £100 ticket. It need not be referred to as a "tax" merely another one of the seemingly never ending list of ancillary charges that appear on tickets already.
The principal would be however that the carrier paid the tax, not the passenger.
And, the charge should be paid by all commercial and business flights, using multi engine aircraft with in excess of 9 seats, or in the case of cargo, say a payload capacity of 1000 kg / 6 cubic metres.
In doing this carriers would be be forced to look at their loads and remove excess capacity, where it exists, to reduce costs / maximise profitability and have the alleged desired result of operating the tax as an environmental tax, rather than another way of generating more money for government koffers.