No. They cannot give permission to search stuff that belongs to TS or her daughter. And it would be illogical to suggest that CSI did not search such belongings. We know that TS and her daughter were denied access to their stuff per TS who stated that they had to buy underwear etc because they were kicked out and had no clothes.
So Al Stauch giving LE permission to search his home would NOT hold up in court since TS lived there.
Further, even if he gave permission and TS hadn’t lived there, procedure for LE when conducting a major search without emergency and the seizure of evidence in that manner, is to get a search warrant. Period.