That sounds very reasonable. With the current method, an inter-site read has to be made every time somebody loads stunts.hu, while with what you suggest, the slowdown would only occur twice a month, when a race is started or ends in Race For Kicks.
I have thoughts of ideas similar to this before, but I haven't yet done it. So far, I'm familiar with sending a POST when the user presses a Submit button, but I don't know how to trigger a POST with a PHP function. I would have to read more about that. Then, I suppose I'd be creating an HTTP form with the necessary information to be sent via POST, only instead of it being triggered by a button, it would be unconditionally sent somehow.
Another thing I don't know is how to have website A send a POST to website B without it resulting in website B being loaded on the browser, instead keeping the same page being browsed.
If you can more or less hint me on these things, I can start to make a picture of what it would be like and I'm sure I can do it.