iPhones are lego projects. There is no reason why you can't simply follow the iFixit guides already here to tear down both phones and build one good one. Yes a 5c will work with board A, battery B, screen A, housing B, etc. all are just legos.
Re: getting a battery out, use a plastic tool and pry a bit from the edge of the housing not the logic board until you can get some tweezers under there and grab the white battery strip and pull it again. Be careful of metal tools puncturing the battery, although it is fun to watch the explosion.
Ah--just noticed this is a 5c not a 5s. While the 5c may have a bad battery, it is more likely that the 5c also has a bad tristar chip on the logic board. 5c owners tend to use cheap gas station charging cables which do not protect the phone from charger voltage surges which damages the tristar chip on the logic board. Repairable, but you'll either need to take up microsoldering or send it to someone who has the equipment to do board repair on iPhone boards.
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