I bought a bedroom of it about ten years ago. The tech's didn't recognize a problem (to be honest, when it came in, it would have got passed me too!) and cleaned as normal.
Cleaned great, but before they had even finished, it split at the family room and bed room seam. It also had pulled up the tack strip along a wall.
Expensive lesson!
I would not HWE, it can be done but too much moisture is a risk.
I would use some type of low moisture method. Host would be fine if it's not too soiled.
OP would be my first choice.
However you could Host it all and spot clean the pet areas with HWE as long as the pet problem is not too big an area.