I believe your issue is a mismatch between Thermocouple type and measurement board.
Even though K and T Type are 'pretty much the same' it is in fact not.
41 µV/°C versus 43 µV/°C a 7.31% difference which matches your errors.
Wiki...
Type K (chromel – alumel) is the most common general purpose thermocouple with a sensitivity of approximately 41 µV/°C
(chromel positive relative to alumel when the junction temperature is higher than the reference temperature).[10] It is inexpensive, and a wide variety of probes are available in its ?200 °C to +1350 °C / -330 °F to +2460 °F range.
Type T (copper – constantan) thermocouples are suited for measurements in the ?200 to 350 °C range. Often used as a differential measurement since only copper wire touches the probes. Since both conductors are non-magnetic, there is no Curie point and thus no abrupt change in characteristics. Type T thermocouples have a sensitivity of about 43 µV/°C
PS
I too have written a K-Type library for the MAX31855...
Thermocouples that are very cheap but good: