If they insist, go ahead and do it. But make sure you fully disclaim any warranty service for WHEN these cables fail.
Alternatively, you could decline the job, stating that you won’t damage your reputation by performing such work. Let them get some college kid that thinks they know cabling do it and they can deal with the fallout.
They do also realize they will be able to feel this right? I know my dad ran speaker wire to his surround sound speakers under the carpet and you can feel this wire, and this is walking on plush residential carpet. If this install is using some thin, commercial carpet, network cable is really going to be noticeable.
My son ran into a similar situation in his home…needed to run cat 6 under the carpet. We put in a flat cat 6 cable just to see if it held up. If it failed after walking on it he would just have a cable run thru the walls but the cost was quoted at ~$500. The flat cat 6 cable was placed under the carpet pad and after 5-6 years it is still going strong!