Your best bet is to try to negotiate directly with the prosecutor. You need to call the court and ask them to put it on a docket where you can speak with the city or county attorneys.
The big problem I see for you is that this is the kind of case they're going to be very upset about because it involves children. If you are the driver and you allow a person under 15 to ride in a vehicle with seat belts without wearing one, then you can be found guilty (Texas Transportation Code 545.413). The ticket for the 15 year old should have been issued to the 15 year old, but the officer was within his rights to issue the other 4 tickets. The maximum fine for this offense is $200, so you may convince them to come down by pointing out that Transportation Code provision. Keep in mind that you can also be charged court costs on each ticket.
Your best bet may be to ask for deferred disposition on these so your driving record doesn't get hammered, but I'm not sure that they will offer that given that we're talking about kids in the car.
Disclaimer: This answer is provided as a public service and as a general response to a general question, it is not meant, and should not be relied upon as specific legal advice, nor does it create an attorney-client relationship.
Keeping in mind that the law is not actually "Click It or Ticket", you will be able to make your way through the offenses after extensive research. Retain an attorney to represent you. Although the minimum fine is $100.00 per offense, the seatbelt offenses for the children aged 7-15 will not affect your driving record.