There is a tax on flights out of the UK known as 'Air Passenger Duty'. On aircrafts where there is more than one class of travel, the tax varies depending on the class in which your ticket is booked. If it is the lowest class of travel, you pay a lower rate of duty. On long-haul flight, this is £20. If you are in anything other than the lowest class of travel, you pay the 'standard' rate of £40. If you try to do a dummy booking on the BD website from the UK to a long-haul destination (MAN-ANU, or MAN-ORD, for instance), and you compare what is charged as taxes and fees, you'll see that there is a £20 difference between economy and premium economy. The reason for this is that economy attracts the lower rate of tax whereas premium economy attracts the standard rate.
But don't get too involved in this. I don't think that there is any need to go into tax discussions with BD. If they argue that PE is not a distinct class to economy, I would point out that it has a different seating configuration, a cabin which is physically separated from the economy cabin and that, unlike economy, it attracts the standard rate of Air Passenger Duty rather than the reduced rate and leave it at that.

Are they trying to argue that BA has no equivalent to BD's PE? BA have World Traveler +. Admittedly, most travelers, including me, would argue that WT+ is much nicer than BD's PE, but that is the same as saying that C class on BA is nicer than C class on Cubana. It is still a different cabin. In any event, it is entirely irrelevant. Even if BA had no PE-type product and even if the only seats available on all airlines were plain vanilla economy, you were downgraded. The fact that BD had no other option, assuming this to be the case, is neither here nor there. You are still entitled to the 75% reimbursement, whatever the reason for the cancellation and whatever the reasons for putting you in Y (other than at your own insistence when a PE alternative was offered to you and you declined).

The only complication that could arise is the situation where BD booked you on BA in premium economy (i.e. World Traveler Plus) and BA themselves downgraded you to Y notwithstanding your reservation. That would add a spanner in the works although I still think that BD would be ultimately responsible there. My guess is that, in any event, BD probably got you a seat in ordinary economy on BA.
I don't think that I would be unduly bothered about trying to gather evidence for a future court case at this stage. Presumably you have a copy of your BA boarding pass, and this will indicate in which class you traveled. That should be enough for now.