Hello,
I do not see why you couldn't pipe your console through your recorder and record the feed/channel/video using a RCA splitter. When the feed for the picture comes out of the xbox it does not care what is in the loop as long as the xbox can send the signal to the tv screen to see that is the purpose of the rca video and audio connection.
The thing is the device that records it needs to be able to put it on some form of media and in a format that your computer will understand (so you do not have to mess with converting one format to the one you need although people do this every day).
I am wondering if the console was piped through the DVR or Tivo would it allow you to record and then use the network to access the box and copy the recording. This is just a thought because again , I do not if it would pipe out to your pc a format that you can work with or convert.
A video capture card would also offer a solution but you should research which one will give you the tools, options and resolution you want.
Below is a link that offers you a tested solution using a box in the connection but, it has not been tested with a Wii. It still will give you an Idea on how to do this sort of thing.
As Nick said above the Dazzle might work for you to. I the second link I found explains what Nick was talking about a bit more (in a video format) so he deserves the credit for this idea. This video is really good I think this guy explained this Dazzle well. Give him a good comment for me would ya?
The 3rd link is to the Pinnacle web site where they make the Dazzle device.
The forth link is recognition to the user "RAMSEY" on the teamxbox.com forum. This is where I found the 2nd link's address from where he posted the this video to help another user that wanted to do what you want to do.
Have a good day.