They sound like seperate issues the key might be under a 24month warranty so if the batterys failing they will replace it , I'd suggest checking the voltage of the car battery in the morning before its first start up a healthy battery should read around 12.5-12.6v you can also test the alternator by taking a reading while the cars running should be up around 14v iirc
finally try charging the key on an electric toothbrush charging stand if you have worked a treat on my key