There is no justifiable reason for AA to not pass on the tax (as long as that class based stupidity exists) other than competition. UA isn't, for the moment, doing this for their SWUs.

I endorse the transit through LHR/LGW idea as well and not just for this tax reason.

Note another stupidity of the UK law. If you fly from UK to the US via any European city (CDG/BRU/ZRH etc.) as a connection and upgrade the transatlantic part, you will still need to pay the UK government the additional 20 quid. Don't know if AA is collecting this as well.