You're not working on an Airplane. I'm assuming you're not stepping on or into the the aircraft itself. In fact your general liability policy excludes that kind of coverage; no news there. The "other" cleaners with the huge liability premium are probably aircraft detailers for the most part and dabble in cleaning exteriors and interiors. They don't know shit about aircraft maintenance or how to properly clean textiles, generally. They are always required by the aircraft repair facility and/or owner to have that coverage. They are physically working on and in the aircraft.
You can work in the aircraft hanger on "carpet" that they provide you in a space that they allow you to work in and not touch the aircraft. Hell, you could clean it in the parking lot or take it to a shop and avoid the liability of touching the airplane. If you damage or destroy the carpet you could be held liable but unfortunately you're insurance provider would deny any claim once they determined it was carpet from an aircraft.
In a previous life when I cleaned aircraft carpet I cleaned it in an area away from the plane, and the shop (repair station authorization) indemified me from any damage to the carpet because I was in fact not cleaning it in accordance with the approved repair manual proceedures (Host Sponges). My invoice said "Carpet Cleaning Service" and bam I got paid.
Personally I think you have (or maybe the opportunity is gone) an excellent chance at restoring the carpet with KleetRite All Dry Solution, RTU at $33/gal. If the aircrafts maintenance manual is deferring to the carpet manufacturers cleaning instructions then you're halfway there. Clean a small area, present results to the owner or maintenance manager and tell them you'll clean "carpet" as provided with an invoice with minimal information and the owner or shop assumes all liability to the end of time for any damage from the cleaning process. You're not going to damage the carpet and if there is a fire retard treatment on the carpet backing you're not going to effect that either. They get clean carpet without replacing it and you get paid a pile of money. If they won't indemnify you then they can go f... themselves and replace the carpet.
