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.
Since your device is a intentional radiator, you need to be very conservative. Common sense may say some changes don't matter, but that's a judgement call the paper pushers get to make and it's very expensive to argue with them. To be fair, some little seemingly inconsequetial changes can effect emissions in unexpected ways.
To be really safe, test your final product and then ship only that.
Get cracking on prototyping with the low resolution DAC so you know which one you want as soon as possible.
Best Answer
This falls under the topic of "permissive changes". The FCC has a document called PERMISSIVE CHANGE POLICY which outlines what can or cannot be changed without requiring a new FCC ID.
Under section 2 c). it says, "Part substitution – electrically identical parts may be substituted."
I would assume that the two batteries are the same voltage, but perhaps different capacities (mAh) or chemistries? For example, allowing for either rechargeable or non-rechargeable batteries in the same package (such as two AA's) shouldn't be a problem. However if the two battery variants are substantially different in size, it could be argued that would affect the RF performance of the device (since they have a metal casing) and require re-testing.
Your RF testing house may be able to give you some guidance on whether different size batteries would affect the results.