Possibly the simpler solution would be:
Either stop allowing tourists and normal citizens in the building or
Build another Home where the President and First Family actually lives
Isn't there some free space on the lawn around there?
The people visiting is not the problem because of a series of checkpoints & searches. The issue is/was this person breached security by jumping the fence, which is not a small feat. This has happened before and either they were captured as soon as they scaled the fence or a short distance afterward when the dog(s) caught them.
The President and the First Family live on the second floor. The White House is close to if not in the center of the complex. So if there was a separate home built, it would be closer to the perimeter fence and there would be an issue with security. Plus it's easier to maintain a high level of security for one area instead of two. If additional support is needed, you can focus it on a specific area.
There is a buffer zone around the complex since the streets are closed except for foot traffic. Even that has some restrictions when the guy shot at the White House with an AK-47 some years ago.
This reminds me of when a small plane crashed on the White House grounds.