Action Congress can definitely take to deal with amnesty.

The Constitution’s only specification with regards to the State of the Union address is:

He shall from time to time give to Congress information of the State of the Union and recommend to their Consideration such measures as he shall judge necessary and expedient.

Until 100 years ago, this was generally a written report.  Then President Wilson gave an address in person, and it has been done that way ever since.  However, if Congress does not invite the President to appear, then the State of the Union cannot be delivered to a joint session.  The President would whine about it, especially as he loves to deliver speeches, but there is nothing he can do to force the issue.

If asked, Republicans can reply that the President is supposed to faithfully execute the laws.