In the x220 battery packs the controllers are not locked down hard (the later x230 batteries appear locked down and cannot be reset the same way... likely need either special software or may be able to connect directly to flash). The controller (bq8030 or bq2090 or similar) nvram can be dumped and written using the laptops built-in battery management smbus.
I recently re-celled mine; dump the calibration nvram before starting (not the end of the world if you didn't). Buy whatever power tool battery is on clearance. Swap the 18650 li-ion cells. For a cheap and cheerful spot welder, you can buy a board with some FETs to switch power from a car battery. Restore the firmware. Run Lenovo battery calibration.
Lots of info at http://www.karosium.com/2016/08/hacking-bq8030-with-sanyo-fi... (no affiliation, but found it a great resource!). If you didn't save your firmware or starting from a toasted battery, compare and contrast to calibration dumps found online. TI also has general specs available on a similar controller.
The battery controller will set a bit to disable the battery when the cell voltages get out of sync when removing/adding cells (other ways of dealing with this too.... holding controller in reset, voltage in parallel). Restoring the nvram calibration data resets this. Finding most data in the dump was relatively easy by comparing to battery statistics available through ThinkPad tools. I have detailed notes and posted to comments at above site.
There's some debate on different cell charging profiles as the original cells were 4.1v versus modern chemistry being 4.2v and so potentially will end up undercharged... but I've had great success just dropping in and don't feel the need to squeeze out more capacity.
Took an afternoon to do....not necessarily worth it, but was a fun project to figure out. My 9-cell is back to 10 hours of runtime/way better than any aftermarket I've tried.