I am having a similar issue after upgrading to Big Sur. Battery shows Service Recommended and shuts down as soon as I unplug it. When plugged in, the battery won't charge and the power source is the power adapter. My battery cycle count is 132.
MacBook Pro (13-inch, 2016, Two Thunderbolt 3 ports)
2 GHz Dual-Core Intel Core i5, 8 GB ram, Intel Iris Graphics 540 1536 MB