Here's the deal. Yes a new strong battery is essential and being regularly on a Battery Tender. I went through the same thing with mine. Low voltage, or next to no voltage makes them go
"Nuts"!
Then, being an American, I finally decided to read The Manual about fob reprogramming.
Open the little flap on the bumper once the battery was fully recharged. Turn the key 'X' number of times as instructed while having the fob in the little pocket in the glove box facing to the right. Walla! Everything worked again!
Yet, even after doing this it will still occasionally lose the Fob Signal. This particularly when absent from the car for a few days.
On the left side inside of the glove box is a little pocket. If the display says "No Fob Found" (even after reprogramming) simply stick the Fob in the little pocket in the glove box
with the buttons facing to the Right Passenger Side. Don't worry about the back and forth deal in the trunk access little square plate. The car (assuming a fully charged battery) will then start and will then re-learn it's default settings to the Fob. Whenever this may occur (again, assuming a fully charged good battery), just use this process. For immediate and relatively continuous use the car will then re-learn to recognize that Fob. Will you have to repeat this process every time? No. Just once in awhile if the Fob is absent from the car and not used for several days.
Don't forget, the Fobs also have little batteries in them on any car thus equipped that can and do go bad.
Hope this helps.