Need accuracy of 8-bit ladder to 1/2 LSB (9.75mV). 10-bit ADC precision will not help if the accuracy of the 8-bit ladder is beyond 0.5 LSB as there'll be missing 8-bit codes.
The MSB resistor effects the output voltage by 5/2 = 2.500V. For ±1% tolerance, the effect is ±25mV = 2.5 LSB
Need to pick MSB resistor to achieve ±0.25LSB (0.5 LSB range) which is ±4.88mV. Therefore, the MSB resistor tolerance needs to be better than ±0.2%
I suggest testing and recording all 8-bit readings. Then calculate step change (in LSB) and error (%) for each reading.