Personally, and this might not be universally popular here, I would treat with a glysophate compound and wait the recommended period. If the carpet backing is synthetic(polypropylene) then it must be removed as its toxic and a total menace to any form of cultivation. If by good fortune the backing is jute(natural string) then its probably already decaying and could be left in a raised bed, although you might struggle to plant through it. Once the weeds are dead, you may be able to prise off enough , or to plant plants through it for many crops. Seed beds upon the topsoil if buried deeply.
Alternatively, if you can muster the muscle power, (rugby club friends? ) armed with a scaffold pole, wrap sufficient carpet along to get purchase and then (simply!) heave. Do not expect to get it off for Ebay!
Remember, Rome was not built in a day (if you've been there, you will know they have not yet finished). Plant whatever you can, work on the remainder piecemeal and get a scheme in place. Not all your crops are planted at once.