With today’s technology it should not be a problem to move the entire press room to another building. Then set up desk with microphones (controlled by the White House).
The President could appear on their screen and he could see them. He would then control the microphone of the reporter.
Or...simply give up the daily briefing
Set up a web site where everyone has access and post the daily briefings on the web site.
Such a site already exists and has for awhile.
click link above...bookmark it, get the real skinny daily