His mom is right.
It's not necessarily uncaring, it is rather immature though and borderline deceptive (since they wont know you're already married).
You can have only 1 wedding per marriage. A wedding is a legal ceremony for 2 unwed people to marry. Whether you get married in a courthouse, church, beach, your back yard, etc as long as you have fulfilled the legal requirements, you're just as married each way. A courthouse ceremony ... ***is a wedding***.
If you are legally married on the 13th and have a fake wedding on the 17th (yup, I said it... fake wedding), it's not going to make you more married.
Your reception is your thank you gift to your guests for attending your wedding. Why would they want to come to a reception, toting gifts for an event that has already taken place, which you will be excluding them from? They are not attending your wedding, just the re-enactment. Also, if you marry on the 13th, your reception on the 17th, isn't even a reception. A reception is something held the same day as the event. The concept of a reception, is to receive your guests following your event. They are attending a play.
Have you spoken with your officiant for the 17th on this? If it's a clergy, they may not be interested or willing to participate in a fake wedding. They are who turns the marriage license back into the courthouse so they will clearly know you're already married.
Sometimes you have to pull up the big girl pants and realize sometimes you have to be practical. The day of your wedding will be special to you no matter what the date is, even if it's not an "anniversary".