I've changed addresses three times since receiving my Texas CHL. Each time I did the work for DL and CHL address changes online and never worried about carrying in the interim. If a LEO here runs your DL, I understand that your CHL status is received in the response. Assuming you're acting correctly and the LEO involved is reasonable, there should be no problems.
I printed the webpage showing I had applied for the address change and carried it in the console just in case.