I tried it on another save and it worked, but there were a couple variables that may or may not have contributed to it working.
First of all, it wasn't the same house. Since you had asked me to try it on a different save, I just started a new game and immediately went to buy the house. Since I didn't take the time to get a job or earn money, I couldn't afford the same house that's giving me the problem.
Second of all, in the save that's giving me issues, I ran out of free space because prior to attempting to build a house on an empty lot, I lived in the mansion-style house. In that house, there is no free space available probably due to the size of the house and the amount of furniture in it. Even after selling some of the furniture in that house, though, there was STILL no free space available. That was why I wanted to move in the first place. Therefore, it might be a problem from the free space issue with the other house that's somehow affecting me building a house in an empty lot as well. In the new save that I started, I obviously didn't live ANYWHERE before I attempted to build the house on the empty lot, so I never had a free space issue before attempting to do so.
I hope this all makes sense. I'm not sure if any of that makes a difference or not, but I just wanted to state those variables just in case!