From an SEO standpoint, I don’t think it’s a good idea to keep changing the actual home page (and its URL every three months). Also, people with bookmarks to the site won’t necessarily have them point to the right place. It would be much better, IMO, to change the *content* of the home page four times a year, without changing site_start, the pages URL, or its ID.
How about setting things up so the page changes automatically on the appropriate dates?
There are about 10 different ways to do that in MODx.
I recently posted a "season" snippet around here somewhere that loads one of four different chunks depending on the season. That would be one way.
You could also use a template variable to set the season and use that TV as a parameter to a snippet that produces the page output.
Or the TV could let you select the appropriate chunk to insert as the page content.
Or you could use a snippet on the home page with $modx->sendRedirect to forward the user to the desired page.
etc., etc.