Thanks for the replies. I was hoping someone has experienced the same and might be able to give us some advice.
I should say that we have made every effort to address the underlying issue. Our vet has ruled out any physical reasons. As I stated, it has been going on for well over 3 months with no end in sight, so entertaining the thought of giving him up is no rash decision.
Oscar's litter box is clean (this missed day or two is a rare occurence). We have two large litter boxes for them both. He gets plenty of food and water. In fact, he was on a diet prior to my wife giving birth with rationed and monitored feeding. However, we've scrapped the idea in an effort to appease him thinking he was protesting the lack of readily available food. Our other cat Milo is perfectly fine and there are no other cats or animals in our neighborhood that would present a threat. BTW, both cats are completely indoor 100% of the time and get along just fine.
I'll try the enzyme cleaner, but Oscar goes in different locations throughout the house. The only constant area is the front door mat. I removed the mat for a while to see if it made a difference, but he still went any way. We're at a loss for what to do next.