I would say correction. That you've purchased a Business Class product on ba.com for a sector operated by a oneworld code share partner. Therefor not a Club WORLD service for all that entails, all be it another IAG product.

I'm guessing from your other comments that this routing was chosen to secure GGL. Had you flown AA domestic to connect to BA to LHR instead of via MAD then you wouldn't have had issue as you'd have been arriving on the eligible product for access.

If a shower is so important, then why didn't you factor in a longer connection at MAD to make use of the lounge facilities there?

I dont see this as being down to interpretation on any level. The entry requirements are quite specific and clear. Unfortunately in your situation they appear not to apply to the travel itinerary for which you have booked yourself.