Progress must have been made in the last months since you wrote that post, because Kaeru's WFC is working perfectly on my end using the latest Nightly of Desmume.
The procedure is very simple, I'll write it very quickly here for anyone who's interested.
1) Download the latest Stable version of Desmume.
You can find it right here.
2) Download the latest Nightly version of Desmume.
You can find it right here.
3) Download the latest installer of WinPcap.
You can find it right here.
4) Configure Desmume as you see fit, loading your BIOS7, BIOS9 and Firmware in the process. You can do that under Config > Emulation Settings.
5) Go to Config > Wifi Settings and make sure that "Enable WiFi Emulation" is checked and the Bridge Network Adapter selected is your main Ethernet Controller.
6) Boot up your game and go to its Nintendo WFC Settings.
7) Delete all the Connection Settings there or modify the Connection 1 directly, whatever floats your boat.
8) Make sure to do the following modifications:
* Set "Auto-obtain DNS" to "No".
* Set the Primary DNS to "164.132.44.106"
* Set the Secondary DNS to "1.1.1.1"
9) Save Settings
10) Go back to the Connection 1 Settings and run the Test Connection function, to make sure that everything is working correctly.
And that's it, really. Go back to the main menu of your game, and you should be able to access its WiFi functions normally.
Notes:
* The latest Nightly version is always shared as a single .exe file, which won't work on its own. That's why I'm asking to download the latest Stable version of Desmume first, so you can drag and drop the latest Nightly in your Desmume folder.
* You must be connected to the internet via Ethernet connection. WiFi connection will not work, if I recall correctly.
* Unlike the main post of this thread says, setting the Secondary DNS to the same DNS Address as the Primary DNS got me locked to one single Mystery Gift in the Pokémon Games with Kaeru's WFC. When I changed the Secondary DNS to 1.1.1.1, I was able to get other Mystery Gifts as well however.
EDIT: This might have actually no effect at all. I booted up Platinum just now and tried to get another Mystery Gift. At first I got a Shiny Milotic, but since I had 3 Wonder Cards, I had to go back, delete 1 and connect to "Nintendo's" WFC again. However, on that 2nd attempt, what I found was a Wonder Card that I already had.
I guess this is fully random. Sometimes you get a new Mystery Gift, and sometimes you get a repeated one.
I'd personally still suggest to use 1.1.1.1 though, because that's what Team Kaeru suggests in their official website.
If anyone sees anything wrong with this post, let me know and I'll fix it up.