1. I can suggest you to use an OPAMP with very low input offset voltage (also known as Chopper or auto zero or zero drift. the name varies with manufacturer but the functions are same). Non of the OPAMP which is embedded into the microcontroller will suitable to interface your sensor (except Cypress PSoC controller wich is very much expensive compared to the external OPAMP design). I will never suggest you to use the internal OPAMP of a microcontroller.
2. As you mentioned, 8 bit ADC is not enough for your requirement (in current days microcontrollers are comes with 10 bit at the least), but 10 bit or 12 bit Embedded microcontroller ADC is good enough with a good OPAMP design.
3. Cost: if your are looking for low cost microcontroller with
high volume production, I can suggest you many (specially ST microcontroller) and it is worth to learn and write a program for the specific controller. If the requirement is low (below 100), it is better you to stick with whichever controller you are familier with. Example Microchip PIC16 series.
For very low cost OPAMP you can consider OP07 and LMV358 (see the specification) or you can use the link below for zero drift OPAMPs
https://www.digikey.com/product-sea...=1&stock=1&quantity=0&ptm=0&fid=0&pageSize=25
Udhay