Concerning the fob battery, I highly recommend buying a quality Alkaline battery for it from a reputable supplier.
I've bought Duracell from eBay before, only to find out they were cheap crap knock-offs and not Duracell at all.
They are lithium, not alkaline, but I know what you mean. I, too, got ripped off with some fake batteries from Amazon at one point.
Pissed yesterday, because my entire 200 mile trip, most at night, with that damn red light glaring me in my face and I didn't have anything to put over the light. On the dark backroads, it is very annoying and makes it harder to search for debris, deer, etc (why the hell is it so bright???). I already covered the high-beam indicator with a tint sticker many years ago. I need to do the same with the warning light.
I also use the CR2032 as it lasts considerably longer than the CR2025. When you fit it, you need to push it in firmly before snapping the 2 halves back together.
That got me curious. So I took it apart. The battery was 3.0V, which is certainly not that low. But since I had it open, I went ahead and tried to force a new 2032 in there (which I do have plenty of) instead of a 2025. It is a tight fit, but it DOES WORK. That is what they SHOULD have picked in the first place. Generally, a 2032 will have 36% more energy storage than a 2025. So one might expect it to last about 1/3 longer. Anyway, the new battery was 3.3V.
The last time I changed it was only a year ago, and it measured 2.9V at that time, with the new one being 3.3V. Really, it should be fine even as low as 2.8V. They normally last about 2 years.
So I placed the phone on the seat and turned the key- no more low battery message. But could be temperature, could be something else. I will key an eye on it.
Good tip.