Budget time for sleep. Period. It might take a bit of strength to commit to that based on whatever situation you are in, but it just takes practice.
I have been in situations where I was absolutely forced to be awake for about 5’ish days but nothing else in my life after that ever forced that requirement. All-nighters were always a decision I made willingly and they never really worked out well for productivity.
Still, I am a shitty sleeper and at this very moment, I only have time to finish this comment and get about 4 hours sleep in a strange hotel before I need to wake up for work. However, I am older now and less sleep seems to be the norm for my body.
Managing being exhausted is a personal thing. Aside from drinking more water or coffee, I don’t think there are going to be many suggestions that will really help more than that.
If it’s more formal, you don’t say anything. In almost all cases where someone would be tempted to lie, I should already know what the correct information is before I even asked a question.
In less formal situations, I would just keep asking follow-up questions. Lies are generally very shallow.
Edit: My point is that there are methods to call someone out without actually making direct accusations. Accusations are “hostile” and not generally worth derailing a meeting for something that can be dealt with later.