Like those below, I'm not a lawyer, I'm a real estate agent. However, I own a number of rental properties in the Phoenix Area. In every case, they were purchased in either my name or my wife's name, and then later they were transferred over to be titled in the name of one of our LLCs. Recording that transfer will not trigger the due on sale clause, as no sale has happened.
As long as you own the LLC (which can be verified with the corporation commission, as the LLC is also recorded), it's pretty clear you are just changing the way in which you hold title. It would be the same (non-) event, if you transferred your primary residence from your family trust to yourself in your own name or vice-versa.