I changed batteries in both my fobs without any problems at all. I doubt you lose programming when simply changing a battery or a lot of us would have run into the problem. It’s possible you put a dead battery in there. That’s unlikely, though, but you do have something is going on so it’s the first thing I’d rule out.
There’s a device called a multimeter that’s well worth having. They can be as cheap as $5 up to over $100. Checking batteries is one thing they do well but a multimeter is something useful for many things. For the very cheap, try Harbor Freight Tools. Otherwise Amazon has them. I like the auto-off feature because II tend to leave them on. Besides checking batteries, you can use a multimeter to check circuits, check to see if a wire makes a connection (as in a wire that produces a short circuit), or whether a wire is broken, you can check to see if power is supplied to a wire with the car off, for installing something (like a dash cam). You can check fuses with them, old filament style light bulbs. There are some multimeters built with a check batteries feature, they put a small load on the battery then indicate whether the voltage is maintained with the load as with a good battery, or whether the voltage drops with a load, as with a battery that is near dead but still has some voltage.
For the fob, a new battery is worth trying. If you have one working fob, that should keep you going until you find yourself near a service center. Get a name brand battery, occasionally some of the really cheap batteries can leak. That can put a nonconductive surface on the battery contacts. So while you have the fob open make sure the battery contacts are clean and shiny.
I believe they have new fobs, ones that avoid the theft risk of code copying by thieves. I don’t know if the 2013 models are compatible with the new fobs but if you’re going to spend $87, it might be worth looking into the new one, can’t say. I’m not in a high thievery area. Still it’s all or nothing, you only need one thief.
As I recall there’s a place you can put a fob to start the car even with a dead fob battery, I think it is near my console somewhere, but that might tell you if your fob still works for the car. I’m guessing here but it should be easy to try.
So my suggestions:
Try another new fob battery
Try fob in “dead fob battery” area of car to see if fob works there
Enable smart phone use for opening, locking, and driving car if you have a smart phone capable.
Get an inexpensive multimeter if you are a do it yourself sort of a person
Please let us know what the problem was once you sort it out.
