The mileage is kept in non volatile memory. If your clock is dead, and it was being kept by the same battery, common sense would dictate that the mileage would be dead as well right? It's actually a law so it is kept non vol so that it can't be toyed with very easily to keep folks honest.
If the battery is soldered on, a lot are just so they don't shake loose or get jarred and cause issues, just unsolder the two points and put a new one on it's not that hard. Use a point iron don't go at it with a 100 watt monster tip or a squeeze and squeege instant on iron.
If you really don't like that, unsolder the battery and get a battery like a 123 or something that fits into a holder, solder the holder back in place of the battery (watch the polarity so it's correct) you now have a replaceable battery. You can tape it or velcro it or silicone it in place so it don't bounce around. You can also get holders for the 2032 and likes lith battery buttons that have two wires on them, and use that so you can put the original size battery back in if there are space limitations.
It's might be a long shot depending on where you live, but look for old style electronic repair shops, generally you'll find an old hammie around

who would be happy to help you out with that, explain they are only replacing the battery and don't worry about code or chips and crap and they'll be like, oh okay i can do that !! I do repairs for folks from time to time on stuff, if you want me to 2m / 3m it, no guarantees but little stuff like that, . a lot of people can do.
Aaron