Bluetooth low energy digital thermometer approvals
My understanding is that if your digital device is capable of plugging into mains power, you will have to at least get unintentional radiator testing done. From what I hear (no personal experience, just research), testing for unintentional radiators is relatively cheap and easy. The sparkfun link below estimates the cost at about $1500.

For intentional radiators, it seems that using a certified device with modular approval, subject to the module's grant restrictions, allows you to use the certification of the module.  You still have to have the unintentional radiator testing done though.

See:

http://www.rfi-global.com/images/uploads/file/FCC%20Certification%20-%20Overview%20For%20Integrated%20Wireless%20Devices.pdf

http://www.digikey.com/en-US/articles/techzone/2011/mar/the-fcc-road-part-15-from-concept-to-approval

https://www.sparkfun.com/tutorials/398