iPhone claiming low battery in cold weather
I would question two things about your framing of the issue (and will sidestep the whole repair discussion/issue entirely):

1) Apple's warranty does cover batteries that don't retain a percentage of their useful life based on aging at one, two and three years (two for most iOS warranty / coverage, but there are exceptions in some cases). You should open a ticket with them if you have coverage and see if they will repair it for you.

2) I use an iPhone regularly outdoors when the temperature reaches -30 C and even then, the decrease in capacity is temporary. While the device is colder than 0C - you get less juice out of the cell, but once you warm it (in a pocket, next to your skin to warm it temporarily - or when you are inside a heated space for long enough to warm the battery), the devices recover full capacity. 

(You will want to watch to control condensation to prevent liquid damage / especially if you bring a very cold device into a warm humid space, that warm air will carry liquid inside the product. Similarly, a warm device when rapidly cooled could have liquid condense inside it and then freeze.) Devices left in a car overnight such as iPad and Mac also see no measurable decrease in battery life once I get them inside and warm - whether I charge them or not.

I'll leave the DIY repair / pay for a repair discussion for another thread where you get specific on what model, but from what I see, the environmental behavior is that cold batteries still have all the charge they took, but they don't produce as much voltage and current while they are cold.