You can try running the third party app EtreCheck and posting the complete report here so we can examine it for possible clues.
How to use the Add Text Feature When Posting an EtreCheck Report - Apple Community
Tahoe does have a lot of visually related bugs as well as some other system related bugs which can affect memory & storage. I recommend rebooting Tahoe about once a week, or every other week at least until Apple sorts out the issues.
FYI, the quality of most third party Lithium batteries is extremely poor. The only vendors which tend to have better odds are OWC and iFixIt (just make sure to calibrate/exercise the battery on first use as directed to minimize the chance of issues). Also, this forum is full of posts where people damaged their laptop while attempting to replace the battery where the laptop was no longer usable.
The crashes would not be caused by a bad battery. That is either some other hardware issue, or an issue with some third party software that is installed.