I don't know about people performing this as a service, but it's easy enough to do... sort of.
I thought I had put together a webpage years ago, but I can't seem to find it so I guess I never did.
Anyway, the front plastic 'bulge' of the Ten no Koe Bank is just held on with double-sided sticky tape of some kind. And this is the most perilous part - because I couldn't find similar sticky tape, so I couldn't get the lid to fit back on perfectly (but it was good enough for me).
The battery inside is a CR2016, but at the time I could only find a CR2032. Electrically, it's the same (higher-capacity actually), but it's a bit thicker, and may have had something to do with the plastic front not being as flush as before.
The battery is soldered (or was it spot-welded ?) onto a couple of contacts which connect to the board; this is also a bit of a tricky spot, since you would need to solder the battery in, and this type of battery isn't supposed to be exposed to that kind of heat. Mine didn't explode, but I was paranoid about how long the battery was exposed to the heat. If you can find an ultra-low-profile socket, more power to you. They're likely easier to find today than when I did my Ten no Koe surgery.
Once the battery is connected up again, put the plastic cover back on with new double-sided sticky tape. Probably somebody around here could put together a nice tutorial...