That is the best option.
Wasn't there a case like this...?
Guy tried to make an appointment to get his LTC renewed because it was due to expire soon. He was given an appointment AFTER the LTC expired. Upon entering the police station he was promptly arrested for possessing guns without a valid LTC.
In your example, I'm sure it will be pointed out that he would be ok as long as he "applied" before the old LTC expired regardless of the appointment. BUT, I don't have the confidence others do in all LE (some yes, but not all) doing the right thing and I can see them charging someone like this. Of course they (the state) will lose in court, as long as the accused can afford a good attorney. And and this is the basis of my "paranoid" over protection attitude. When the accusation can destroy your life financially, are you better off being careful and right, or just being right.