I can't speak for Texas but most States aren't going to mandate much in the way of how a prisoner is guarded at a medical facility with a few exceptions. For the most part an officer is going to allow medical staff to do what they need to do. The only issues is a few Dr's or other staff members that don't understand that many times it was a fight to get people into cuffs and I'm damn sure not going to release them without a very compelling reason.
Interference with a Law Enforcement official may result in obstruction charges. Hospital Policy doesn't trump Law Enforecements authority to conduct business.
I have had a few Dr's that "ordered" me to remove a cuff and I simply explain that it took several of us to get him in those cuffs and I'm not releasing them. If they insist I explain that if the prisoner attacks them its on them. I also explain that if an escape attempt is made they will be charged and going to jail for aiding escape. Simple, they understand the consequences and take the liability for their own poor choices. It keeps it to medically necessary. Usually its not an issue. People being taken from a detention facility would be under similar guidelines. But they may or may not be restrained depending on their classification in the detention facility.
The early bird may get the worm, but the second mouse gets the cheese.............