Hey Ashley,
I haven't heard of anyone doing that, but it seems like a good idea. They'll pay for their own printing, and you just have to factor in the time it'll take to stuff 1000 newsletters...unless your printing company is willing to do it for you.
The only downside I see is that your readers may be annoyed by it...and if it creates a lot of extra clutter, the distribution locations will be upset.

But, I think it would be worth a trial run to see how it all goes.
Price? Plan enough to cover your time and a reasonable hourly rate (if you are stuffing yourself) - ie. 2 hours @ $30/hour - $60/week. If the printers will do it for you, charge enough to cover their bill and a hefty markup. Just be sure to plan it out and consider all factors before quoting them a price.
Good luck! And let us know what happens.
Ryan