If I (as a website owner) receive a DMCA Takedown Notice for a user's post, am I allowed to give said user a small period of time to provide a counter-notice before I take down the post? I was hoping to give users the ability to fight false takedowns before their content is removed.
The whole point of the DMCA is that you don't get involved in any way with the argument, because that means you cannot get sued. You say "none of my business, I take it down if I get a takedown notice, I put it back up if I get a counter notice", and that keeps you out of trouble.
If you get involved, then you can get sued. You say you want to give users the ability to fight false takedowns. At the same time this means that illegal content with a correct DMCA takedown notice stays up longer than it should, so should someone decide to sue for copyright infringement, you are on the hook as well.
Of course you are free to inform the user as soon as possible, send them a form how to submit a counter notice, and when you remove the content, arrange things so that it can be restored very easily, minimising the time that the content is gone.
Even if you did, the material would still have to come down. According to the DMCA, a provider only has immunity if he
replaces the removed material and ceases disabling access to it not less than 10, nor more than 14, business days following receipt of the counter notice, unless its designated agent first receives notice from the person who submitted the notification under subsection (c)(1)(C) that such person has filed an action seeking a court order to restrain the subscriber from engaging in infringing activity relating to the material on the service provider’s system or network."
You might be able to wait a day to give the person time to file a counter-notice, but this time frame still applies. If you waited out the entire 10-14 day period before taking it down, I don't think any court would find that you acted "expeditiously" to remove the allegedly infringing material.
It's not actually illegal to do this; you just lose your immunity. But most people wouldn't want to incur liability if they can avoid it.