I'm no lawyer, but have been thru the FCC testing process a few times. For a ordinary device that doesn't deliberately transmit (called "unintentional radiator" by the FCC), there is no legal requirement for certifcation. There are legal requirements for what it is allowed to emit, but it up to you how to make sure your device works within the rules.
You can simply sell a unintentionally radiating device without testing. However, if someone files a complaint and the device is found to exceed the legal radiation limits, you're in deep doodoo. If you had the device tested by a accredited test lab and they determined it was within the limits, your legal case will be much better. The FCC still has the right to force you to withdraw the product and even confiscate every unit out there, but if you can show you followed accepted practices of testing then there will be much less of a issue of punative actions.
Intentional radiators are a different story. You do have to have FCC certification to legally sell one in the United States. When the device is certified, you get a certification ID, and that ID generally has to be indicated somewhere on the outside of the device where others can see it.
In the case of a bluetooth module, most likely the module vendor has gotten the certification for the module. If not, I wouldn't go near it. Even if so though, you are still on the hook for the product as a whole. The module will also be certified with some restrictions, like a specific list of antennas that it is certified with. If you attach a different antenna, for example, the module is no longer certified and you're on your own.
If you're trying to sell a intentionally radiating product, you'd better talk to a expert early in the process. You can wing it a bit with unitnentional radiators, but you really don't want to play games with intentional radiators, even if you're using a certified module that does all the intentional radiating.
It might be a good idea to talk to a testing house. They generally will know all the rules. Just keep in mind they sell testing services, and their answers may a bit biased towards you needing a lot of testing.
Bypassing CE might not be a good idea but you can self-certify and place the CE mark on your product. Self certifying means you do your own testing and document results. If you believe certain tests can be avoided because you are not infringing how a certain module is recommended to be used then that is acceptable.
Low voltage directive and EMC directive are the obvious hurdles but with a little thought and research you should be fine given what you've said about the product you are building.
Alternatively don't mark with the CE symbol and don't make any claims about it. Buyer beware etc.
Here is the UK's government site that is fairly clear on the subject. Here is the list of directives that may or may not apply to your product - if you click on the "info" column against the relevant directive, more information pops-up.
Best Answer
If you're thinking about ISO-9000 that's about the quality of your organization and its processes, not your products. You can have ISO-9000 certification and still produce the worst crap.
There are certification organizations, like Underwriters Laboratories in the US and TüV (Technischer Überwachungsverein) in Germany. Many components won't have certification, though. You'll find it mostly in safety sensitive components, like where isolation from mains is important.
Other organizations, like IEC (International Electrotechnical Committee) write specifications, but don't certify.
A good manufacturer should hand you its own testing procedures for its components upon request, as well as test reports. Those are sometimes included in datasheets.
Some will refer to certification organization standards, like shown in this snapshot of a random Littelfuse datasheet:
Note that the Underwiters Laboratories (the top logo) and PSE (Product Safety Engineering) logos refer to their respective certification, while the CE logo doesn't imply this. (The CE logo means that you claim to comply with CE standards).