How about using a pre certified bluetooth module ? would this avoid having to redo EMC certs ? OR make very much simpler? Of course there may a "cost penalty " for this option ....
Broadly and simplified:
Adding a bluetooth module means that, not only is the device no longer "electrically equivalent" (which means you are re-certing), but you have gone from "unintentional radiator" to "intentional", which means you have a different and/or additional set of standards.
A pre-certified module (as in, it has modular approval) under FCC means you do not have to test the "intentional radiator" part the module PROVIDED you exactly follow the instructions of the module supplier AND you don't do anything "stupid".
Under CE, pre-certified module isn't a thing. It will certainly help you pass from a technical standpoint, but doesn't really let you "short-cut" the testing and paper work.
Hope that makes sense and gives a rough idea.
edit Apologies for rehashing what others have already said. I should have refreshed.
You can see how well the "regulation but no enforcement" regime is working by picking up any disposable vape from the street and looking for the WEEE "do not bin" symbol. It will probably be there.
Then Germany rejected again because the "CE" was obviously in the wrong style, on a generic sticker (I'd warned, but been told "It'll be /fine/."). And German Customs said our slip directing customers to our site for setup and operating instructions was insufficient. This time, they held our units until we sent satisfactory documentation.
Finally, we were able to send, but with after a warning from Customs that a third error would result in permanent bannage and seizure at the border of anything we sent to Germany.
The device is what gets certified, not its parts.