lucy24 - 12:13 pm on Nov 7, 2012 (gmt 0)
I'm thinking that the best and simplest way would be to ... have his thank-you page either open a 5x5 pixel browser window on my domain or send an email to me if the cookie is present.
This bit implies that the OP is on speaking terms with the advertiser, and the advertiser is amenable to helping supply information. Otherwise he wouldn't be talking about having the advertiser's site do anything.
I don't see that it has to be anything more complicated than putting an administrative gif from your site on the other guy's page. Once the request hits your logs, it should be trivial to find or make a script that fishes it out. You don't even need cookies; with a bit of scripting or equivalent, you can pick up any information that's available to the Thank You page and pack it into the gif itself. As in:
= someone from California bought $753 worth of widgets.
= someone in Minnesota may be cheap, but every bit helps.
= someone has figured out how to do their online shopping with scripting turned off, but they can't hide entirely.
I'm just making this up based on stuff I've done in the past. You're either on good terms with your advertiser or you're calling his bluff. Either way you get the information at a cost of just a few bytes back and forth.