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.
If the device is deemed to be information technology equipment (which it may, since it uses Bluetooth) you would be under the '60950' standard - UL 60950, and if you want to cross the border into Canada, CSA C22.2 #60950 (the standards are largely identical, so it makes sense to get both).
Intertek ETL is a US Nationally Recognized Testing Laboratory (NRTL) authorized to test to UL 60950. You could use UL, CSA, TUV, MET or any other NRTL to do the approval - it doesn't have to be Intertek.
The best way to know if the product needs approval is to get an opinion/quotation from an NRTL - they'll tell you what they believe the product needs and how much a certification will cost.
Best Answer
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.