It was over a decade ago, but believe I changed a FSG71M backup battery myself. Open it up and have a look. The memory freqs will be lost if you do it with no power to the radio, but that's better than causing other problems when there is power.
Perhaps the battery might not be easily found.
Otherwise a "new to you" FSG71M isn't an expensive item. Currently two on Segelflug.de at 100 and 110 Euros plus shipping. There may be adapters available to plug and play a newer radio too.
Jim