That's a good point and it's worth discussing.
Personally I'm 51-49 on this. If one option is the lawyer to send the request until deadline and the other 5 minutes after deadline, both options offer something good, but I choose the first one. The second option offers the lawyer a chance to clearly see what happened and to react accordingly,
but that's a little unfair to the Gs. They can do everything right, place the votes in the last minute and then all their work be destroyed by the lawyer, who gets 5 additional minutes to think. I don't like that. I want everyone to have an equal chance. The Gs can vote in the last seconds, but so can the lawyer send a request. If he sees that people are voting in the last minute, he also gets some 20-30 seconds to decide what to do. If he fails, fuck it. But if he's there in the last minute, he can do something and make the game more interesting.
He might make a good or bad decision, but he needs to feel pressure. Once, I was the lawyer and people started voting suddenly for Cuti in the last 5 minutes. I thought the Gs are voting out a C. I saved him. He was a G

It's these fuck ups that make the game more interesting. And the fuck ups are usually connected with pressure. I'd like everyone to feel pressured and that's why I don't choose the option when the lawyer gets additional time to think things through.
P.S: I won't object, though, if you choose the second option. But I'd give the lawyer 5 minutes tops after deadline.