From the story in the story:
For the past year, I and others like me had been working hard to teach [the Paranet] how to defend themselves.
So the story in the story took place no earlier than somewhere around a year after
White Night occurred. Let's say 10-14 months. Assuming the accuracy of the rest of the timeline, that would place the story between
It's My Birthday, Too (February 14, 8 ASF) and
Heorot (October, 8 ASF).
Wild Card takes place a "few months" after
It's My Birthday, Too. So I'd say it takes place between March 1 and June 31 of 8 ASF, at the earliest.
Part of the problem is people usually don't mean 365 days or even 12 months when they say things like "for the past year." Another part of the problem is I can't even guess how long it took between the idea for the paranet and Harry and others like him going out to teach the paranetters. With those two sources of vagueness, we can only guess at the start of the time frame.
The only timing clues I can pick up on in the story encompassing a story are ones that place it after
Proven Guilty. There is also a description of Harry's burned hand that may be useful.
t was a mass of old scars, and not the pretty kind. It had been burned, and badly, several years before. Wizards heal up better than regular folks, over the long term. I could move it again, and I had feeling back in parts of all the fingers.
I recall that Harry was given hope that his hand would work again in
Dead Beat, but I don't recall the progress.
The class takes place between the about a year and a few months after
White Night and a few months before whenever Billy finds Dresden's write up of the class and includes it in the RPG write up (somewhere between
Small Favor and
Turn Coat).
The bogart story probably takes place before
Small Favor, and the class probably takes place shortly after
Small Favor.