Interesting data point: I have been messing with a LiFePO4 battery of my own design. I'm getting errors even without a BMS as part of a test.
Pretty sure that it isn't the existence of a BMS causing the issue. Tesla can pretty easily test if it's a Lead Acid or Lithium battery. Charge it to over 14V, then put a load on it. Any real LA battery will drop below 12.5V right away. A lithium chemistry will stay at >13V even with a bit of load on it. It's not that the BMS cuts out in the "over charge" that tells Tesla it's a non-LA battery, it's the float voltage.
The fact that Tesla's max charge appears to stop at exactly 14.60V seems to indicate they are conscious about LiFePO4 batteries and even careful about them.
If this theory is right, a design that takes care of this would be pretty interesting, as it would need to mess with the voltage at different times (which would be also be a power loss and require new HW). Pretty creative if someone can work around it.