Simcha is fortunate enough to have his own private bedroom. It is fully heated/air conditioned, via conversion of a 13 foot by 9 foot area which used to be a screened in porch. He is happy to stay there, and has absolutely never had an accident. In contrast, every so often he still will mark if allowed to roam alone through the house - so we have put that option off limits.
The one place he has marked the most is a small area of living room carpet which is close to a full length mirrored wall. Somehow, I suspect he sees his reflection in the mirror as a challenger, and will mark to show dominance. So much for the IQ of Chihuahuas.