I am afraid you are being given the run around by AOL.
It is correct that BT require a BT based speed test to be run before they will do anything, but the AOL techs should also know that sometimes for reasons best known to someone else the BT speedtest will just not work on some connections, and a fault CAN be raised but it has to be done manually with some notes about the BT speedtest not working on the connection in question, and a number of other speedtest results are included with the fault report.
We do get it sometimes where it has to be pointed out to BT several times before they get it.
Speak with AOL again, and try to get it escalated to someone withing their tech teams that has any idea what they are talking about, or migrate to another ISP that has some knowledge of how broadband works in the UK.