What might a president do to indicate his or her belief that a portion of the legislation just signed into law is unconstitutional and that he or she intends to disregard it or implement it in some way other than congress intends?

issue a signing statement

Explanation:

Based on the information provided within the question it can be said that in this scenario the president may issue a signing statement. Like mentioned in the question this term refers to a written pronouncement issued by a President of the United States stating that he/she believes that a particular provision would be unconstitutional in many of the applications in regards to the bill about to be signed into law.