Could you cover the cardboard with compost or leaf mould or similar?Straw is cheap, for example. I've used layers of cardboard, compost and straw to clear weedy beds over the course of a year...you can cut slits in cardboard to plant potatoes through, for example.
I'm actually a bit puzzled at their complaints as to the cardboard, actually. Using cardboard as a mulch is a long standing technique...if it was good enough for The Great Geoff H, it should be good enough for anyone. Carpet is a bit different, unless it's 100% wool and even then it can be a bit iffy.
If I were you I would write a politely formal letter of reply to the comittee (keep a copy) repeating your reason as to why the carpet is still there and also repeating that you requested it removed when you took over the plot. Add that you'd be delighted to help load it into and out of a car if only someone would give you a lift to the dump. Also ask them precisely why they object to the cardboard (you may quote me on Geoff) and ask if covering it with other mulch material is an option.
I'd also have a good look at the allotment rules, just to see what discretion the comittee has when it comes to interpreting them.
Finally, if you can't beat the b***rs, join them. There's no surer way to get an allotment site running the way YOU want it than to get on the comittee. (Which is why there's gangs of Ye Olde Boys running allotments their way all over the country.) I went on our committee eight years back to represent Bolshie Semi-Organic Recycling Wimmin on our comittee, and the comittee never fully recovered from the blow of having me there. Half of them have now retired, and tour committee is a far more mellow and benevolent group, belive me. (Except when it came to someone using pink nylon carpet as a mulch...that just HAD to go.)