Answering specific to your question about, "without actively sending customers to your competitors site?", I am not sure if we can get the visitor data from the organic visitors to the original site in any way. So, I guess, a little bit of third party promotion (say via Digg,Stumbleupon) will be required although you will be doing a favor to competition that way.
What we do is we create a copy of the page's source code onto our database and showcase this code to the visitors. This means, although the user will be able to see click-heatmaps from the site, it will not affect the competitor in any way since the functionality of the page will not change in any way.
Also, at the end of 30 days, we stop tracking and will instead redirect the page to the end URL like any URL shortener. I have a feeling that 30 days could be a bit long. If it indeed turns out true, we can reduce that, or maybe provide this as an option to the URL creator.
It would have been nice if they did not have the "minimum $10" clause. I am sure there should be a lot of students and third world people willing to help within their means.