How does an insurance claim work when defective carpetting has been installed?

I have had the worst experience with a construction company restoring my home after water damage. Problems galore. The third time I called about my bad carpet I had someone from a different company look at it. He said it was poor grade, substandard and delaminating. I filed a complaint with the mill through the vendor. The construction company says the mill will replace carpet with the same style.
The carpet, pad and install ect. was totally overcharged. I do not want carpet replaced from the mill, I want different carpet where I pick it out and my own installer. I could get plush carpet and not the sub standard junk that was reported by a carpet specialist as defectivre. I also want a new company to install it. Can I demand this????