After I read the OP earlier, I checked Texas Workers Comp laws online. The law states that workers comp is optional. Where it got interesting was on a non-govt law site that stated Texas work comp laws let the employer choose between providing workers comp insurance or being subject to civil lawsuit. So that sort of explained the "optional" part of the law.
This leads me to believe that what the Hotel wants you to prove, is that you have workers comp insurance for yourself so that hiring you does not make them subject to civil lawsuit if you get hurt on the job.
Just a guess on my part. Curious to know what you find out from the
PR woman.