Briefings should not be about getting into tussles with the press. They are not about scoring points.
No matter what Trump says or does, it will be misreported anyway. He never really wins anything when he engages some reporter in a pissing match.
IMHO, best thing to do is answer their questions as best as he can, then move on to the next question. If there is a reporter giving him a particular hard time, then don't call on him. Don't engage him in any way. This is what his enemies are looking for -- a rope to hang him with. Why give it to them?
I just had the tv on. There was a campaign ad cobbling sound bytes from Trump's speeches and these briefings. No context, no background, just "he said this and isn't it terrible?" When Trump says, for example, that maybe people should be injected with disinfectant, no one watching the ad would get that maybe he was being sarcastic or he was kidding, or whatever he intended by saying what he said. All they are going to hear is "he thinks it's ok to ingest Lysol!" These kinds of ads will continue till the election and every time Trump gets into a tussle at these briefings or makes some offhand remark, his words will be used against him.
What @mystery-ak suggested is sensible. One or two questions, then turn over the floor to Pence. Don't give the enemy ammunition to use against you.
@Applewood great post as usual. I never say enough how much I enjoy reading them and appreciate your contributions here.

Everytime I think I'm figuring stuff out, the rug gets pulled out from under me. You know that feeling where your tipping back in your chair, and you're just about to fall backwards, but at the last second you catch yourself?
I feel like that all the time, now.
One thing that has given me comfort is seeing my President fearlessly engage with the gotcha press, and speak directly to the American people for 2 hours a day. He's hopeful, surrounded by people I trust, and often braggadocios about how well he and Congress, the media, and the Governors were united to defeat COVID-19. All the old argument's against my President were baked into re-election in the previous 3 years. He's not good at extemporaneous remarks, lousy at reading prepared remarks, and unfit for office, but COVID-19 is not baked into the election. When I see President Trump struggle on TV for 2 hours daily I see a man deeply involved and committed to saving American lives, and now American jobs.
Two weeks ago the media was threatening to stop airing the conferences. Have they not won?
President Trump has great pollsters and our President knows every time he hits 50%. If his polls go down, he'll be back. I just fear for the damage of the soon to be epic tweetstorm coming.