Close, JP courts, but thanks for the shout, good recall! As others have mentioned, a sworn complaint must be filed within 2 years of the offense for a misdemeanor criminal offense to not be outside the statute of limitations. But they aren't trying to pursue criminal charges, this is a civil debt (as it states, she can't go to jail or have it go on her driving record, etc.). There are statutes of limitations for civil causes of action as well. I don't know how a court would classify this cause of action, but the general statute of limitations for debt is 4 years in Texas.
"Dealings between the parties" can reset the statute of limitations in debt collection cases. There is argument over whether simple communications count (most courts say no). Definitely any payment counts, which is why debt collectors love the "Pay $20 just to show you're serious about your obligations" tactic since it resets the clock.
The options are: Ignore it, which they will take as an admission of liability, and will create a civil debt which they will potentially attempt to collect, creating hassle, but also potentially will just go away. Or request the hearing as they offer and state that facts and evidence are lost due to the pursuit of this 6 years after the fact, more than triple the misdemeanor statute of limitations for the same conduct, and more than 2 years after a debt can legally be sued for. I have no familiarity with this company or these hearings, so have no idea which path is better.